Diameter of Random Cayley Graph of Z_q
a r X i v :m a t h /0609620v 1 [m a t h .P R ] 21 S e p 2006Diameter of Random Cayley Graph of Z qGideon Amir∗Ori Gurel-Gurevich†February 2,2008AbstractConsider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators.For k fixed,we prove that thediameter of said graph is asymptotically (in q )of order k√∗Weizmann Institute,Rehovot,76100,Israel.gideon.amir@weizmann.ac.il †Weizmann Institute,Rehovot,76100,Israel.ori.gurel-gurevich@weizmann.ac.il1Various aspects,most notably the mixing time,of random random walks on differentfinite groups have been studied.(See[1],[5]for some examples, and[2]which gives a comprehensive survey).The results usually refer to Abelian groups in varying degrees of generality,from the cyclic group Z n up to generalfinite Abelian groups.Roichman([6])notes that the diameter of the random Cayley graph is bounded by a constant times the mixing time,and applies this bound to the case of general groups of order n and k=⌊log a n⌋.The resulting bound is aq⌉}in Z q(where q is prime).The diameter of this Cayley graph is2√q⌋}.Thediameter of this Cayley graph is now q.Put another way,the diameter does not change when adding or removing0from the generating set but the mixing time might change considerably.Another point of notice is the question of symmetric vs.asymmetric generating sets.The results in[3]are for asymmetric generating sets,i.e.S contains just the k randomly chosen generators.We might as well ask about the mixing times and diameters w.r.t.equally to the symmetric case.The proofs below suggest that the behavior of the diameter in the symmetric and asymmetric cases should be the same up to a multiplicative constant of about2k.It should be noted that the asymptotic behavior of both the mixing time and the diameter,both in the symmetric and asymmetric case are the same as in the case of a k dimensional tori of volume q.This is not coincidental, the structure of the Cayley graph of G w.r.t.S is actually that of Z k,modulo some k-dimensional lattice,which contains the lattice of all multiples of q. Perhaps the mixing time results of[3]could be proved in a more elementary manner using that perspective.2The Main ResultsLet G=Z q the cyclic group of order q,a prime number.Let g1,..,g k be random generators chosen uniformly and independently from G.Denote by Diam(q,k)the random variable which is the diameter of the resulting Cayley graph.All asymptotic results are w.r.t.q while k remainsfixed. Obviously,the diameter must be at leastΩ(k√q)in the following sense:Theorem1For every k and everyǫ>0there exists C such that for all prime qP rob(Diam(q,k)>C k√q)>ǫ33Proof of Theorem1For any x∈Z q andias the event i1g1+i2g2+..+i k g k=x(mod q).Let A x= i and Let A= x∈Z q A x.By standard abuse of notation,we will not distinguish between an event and its indicator variable.We ask for which L is P(A)→1?Obviously,for any0,E(A xiA xi andiA xi and i=λi ,A xiA xi)E(A x i∈L kA xi∈L kE(A xq.V ar(A x i)=1q)<1i∈L k V ar(A xi=i,A xq.4Using Chebyshev’s inequalityP (A x=0)=P (B x=0)≤P (|B x−E (B x)|≥E (B x))≤V ar (B x )(L k /q )2=qL kLet B =x ∈Z q A x ,i.e.the number of points covered by a ball of radius L .Fix C >2and let L =k √Lk =1−1C).Since B is never more than q ,P (B >qC.If a ball of radius L covers more than half the graph,then the graph has at most diameter 2L .This is because a Cayley graph is vertex transitive so all balls of radius L have the same volume and therefore any two such balls intersect.Thus we conclude,that given ǫ>0the probability that the diameter ofthe Cayley graph is more than 2kǫq is less than ǫ.4Proof of Theorem 2Fix D <1.Let L =k√iand A 0as in the previous section.As before,ifj are linearly independent over Z q then A 0j areindependent events.Ifj are linearly dependent then A 0jare infact identical.How many distinct events do we have among {A 0i andi =cq we geti 0j 1=i 1j 0.Take allTherefore,among the vectors taken above every two are linearly independent, so the corresponding events are pairwise independent.Given L how many pairs i0,i1<L are coprime?It is a well known result that P rob(i0⊥i1)tends to,and is always greater than6/π2which is biggerthan1/2(see[?]).i∈IA0i∈IE(A0qV ar(A0q (1−1212V ar(X)=Dqq(1−12(1−12(1−12)Since X is nonnegative,from Cauchy-Schwartz inequality we get E(X)2=E(X1X>0)2≤E(X2)E((1X>0)2)=E(X2)P rob(X>0)ThereforeP rob(X>0)≥E(X)242(1−12)≥DFrom lemma4we get that the probability of A0is bounded away from 0regardless of q.Next we shall show that if A0occurs then many differentLemma 5Let C be such that k k C k −1D <1.If A 0occurs thenDiam (q,k )>ki ∈{1..k√ioccurs.Ifj ′differ in amultiple ofj ∈{1..k√j ′such thatj 1g 1+j 2g 2+..+j k g k =j ′1g 1+j ′2g 2+..+j ′k g k (modq )and j ′t ≤k √j ′is less than k (k √Dq ).If this number is less than q then the diameter cannot be k√To warp up the proof,given C take any D <k −k C 1−k and let ǫ=D/3.From lemma 4we conclude that A 0(w.r.t D )occurs with probability at leastǫ,in which case,by lemma 5Diam (q,k )>k√k√would consist of k independent random variables,g1,..,g k,distributed uni-formly in[0,1].Let A d be the set of all points i1g1+..+i k g k(mod1)with i1,..,i k integers less then some d.Let a d be the largest gap in A d,i.e.the)is not[0,1].The asymptotic behavior of a d largestǫfor which A d+(−ǫ2(or rather,its inverse)should be closely related to the asymptotic behavior of Diam(q,k).AcknowledgementsWe would like to thank our advisor,Itai Benjamini,for suggesting this ques-tion and for useful discussions.References[1]N.Alon,Y.Roichman(1994)Random Cayley graphs and expanders.Random Structures and Algorithms,5(2),271-284[2]M.V.Hildebrand(2005)A survey of random random walks onfinitegroups.Probability Surveys2,33-63[3]M.V.Hildebrand(1994)Random walks supported on random points ofZ/n Z.Probability Theory and Related Fields100,191-203[4]D.Wells(1986)The Penguin Dictionary of Curious and InterestingNumbers.Middlesex,England:Penguin Books,28-29[5]D.B.Wilson,Random random walks on Z d2.Probability Theory and Re-lated Fields108(1997),no.4[6]Y.Roichman,On random random walks.Ann.Probab.24(1996),no.2,1001–1011.8。
证据理论方法详解分析
第五章证据理论（Evidence Theory）方法在本章§1，我们将讨论一种被称之为登普斯特-谢弗（Dempster-Shafer）或谢弗-登普斯特（Shafer-Dempster）理论（简称D-S理论或证据理论）的不精确推理方法。
这一理论最初是以登普斯特（Dempster，1967年）的工作为基础的，登普斯特试图用一个概率区间而不是单一概率数值去建模不确定性. 1976年，谢弗（Shafer，1976年）在《证据的数学理论》一书中扩展和改进了登普斯特工作. D-S理论具有好的理论基础。
确定性因子能被证明是D-S 理论的一种特殊情形。
在§2我们将描述一种简化的证据理论模型MET1 . 在§3我们将给出支持有序命题类问题的具有凸函数性质的简化证据理论模型。
围绕证据理论的一些新的研究工作，将在第六章介绍。
§1D-S理论（Dempster-Shafer Theory）●辨别框架（Frames of Discernment）D-S理论假定有一个用大写希腊字母Θ表示的环境（environment），该环境是一个具有互斥和可穷举元素的集合：Θ = { θ1 , θ2 , ⋯, θn }术语环境在集合论中又被称之为论域（the universe of discourse）。
一些论域的例子可以是：Θ = { airliner , bomber , fighter }Θ = { red , green , blue , orange , yellow }Θ = { barn , grass , person , cow , car }注意，上述集合中的元素都是互斥的。
为了简化我们的讨论，假定Θ是一个有限集合。
其元素是诸如时间、距离、速度等连续变量的D-S 环境上的研究工作已经被做。
理解Θ的一种方式是先提出问题，然后进行回答。
假定Θ = { airliner , bomber , fighter }提问1：“这军用飞机是什么？”；答案1：是Θ的子集{ θ2 , θ3 } = { bomber , fighter }提问2：“这民用飞机是什么？”；答案2：是Θ的子集{ θ1} = { airliner }，{ θ1} 是单元素集合。
离散数学英中名词对照表
离散数学英中名词对照表英文Abel categoryAbel group (commutative group) Abel semigroup Abelian groupabsorption property accessibility relation acyclicaddition principleadequate set of connectives adjacentadjacent matrixadjugateadjunctionaffine planealgebraic closed field algebraic element algebraic extensionalphabetalternating groupannihilatorantecedentanti symmetryanti-isomorphismarc setargumentarityarrangement problem associateassociativeassociative algebraassociatorasymmetricatomatomic formulaaugmenting pigeon hole principle augmenting path automorphism automorphism group of graph auxiliary symbol A 离散数学英文—中文名词axiom of choiceaxiom of equalityaxiom of extensionalityaxiom of infinityaxiom of pairsaxiom of regularityaxiom of replacement for the formulaaxiom of the empty setaxiom of unionB balanced imcomplete block designbarber paradoxbase (base 2 exponential function)base (logarithm function to the base 2)Bell numberBernoulli numberBerry paradoxbiconditionalbijection (one-to-one correspondence)bi-mdulebinary relationbinary operationbinary symmetric channel (BSC)binary treebinomial coefficientbinomial theorembinomial transform bipartite graphblockblockblock codeblock designBondy theoremBoolean algebra Boolean expression Boolean functionBoole homomorophism Boole latticeBoolean matrixBoolean productbound occurrencebound variablebounded latticeBruijn theorem Burnside lemmaC cagecancellation property canonical epimorphism Cantor conjecture Cantor diagonal method Cantor paradoxcapacitycardinal number cardinalityCartesion product of graph Catalan numbercatenationCayley graphCayley theoremceiling functioncell (block)centercertain eventchain (walk) characteristic function characteristic of ring characteristic polynomial check digitsChinese postman problem chromatic number chromatic polynomial circuitcirculant graph circumferenceclassclassical completeness classical consistent cliqueclique numberclose with respect to closed termclosureclosure of graphcode elementcode lengthcode wordcoefficientcoimageco-kernalcoloringcoloring problemcombinationcombination numbercombination with repetationcommon divisorcommon factorcommutativecommutative diagramcommutative ringcommutative seimgroupcomparablecompatible withcomplementcomplement elementcomplement of B with respect to A complementary relation complemented latticecomplete bipartite graphcomplete graphcomplete k-partite graphcomplete latticecomplete matchcomplete n-treecompositecomposite operationcomposition (molecular proposition) composition of graph (lexicographic product) compound statementconcatenation (juxtaposition) concatenation graphconditional statement (implication) congruence relationcongruent toconjectureconjunctive normal form connected component connective connectivityconnectivity relation consecutively consequence (conclusion) conservation of flow consistent (non-contradiction) constructive proofcontain (in)contingencycontinuumcontraction of graph contradiction contravariant functor contrapositiveconversecoproductcorankcorresponding universal map countable (uncountable) countably infinite set counter examplecountingcovariant functorcoveringcovering numbercrossing number of graph cosetcotreecutcut edgecut vertexcyclecycle basiscycle matrixcycle rankcycle spacecycle vectorcyclic groupcyclic indexcyclic permutation cyclic semigroupD De Morgan's law decision procedure decoding table deduction theorem degreedegree sequence derivation algebra Descartes product descendant designated truth value deterministic diagonal functor diagonal matrix diameterdigraphdilemmadirect consequence direct limitdirect sumdirected by inclutiondisconnecteddiscrete Fourier transform discrete graph (null graph) disjoint setdisjunctiondisjunctive normal form disjunctive syllogism distancedistance transitive graph distinguished element distributivedistributive lattice divisibilitydivision subringdivison ringdivisor (factor) dodecahedrondomaindual categorydual formdual graphdual principledual statementdummy variableE eccentricityedge chromatic number edge coloringedge connectivityedge coveringedge covering numberedge cutedge setedge-independence number eigenvalue of graph element (entry) elementary divisor ideal elementary product elementary sumempty graphempty relationempty set endomorphismendpointentry (element) enumeration function epimorphismequipotentequivalenceequivalent category equivalent class equivalent matrix equivalent object equivalent relationerror functionerror patternEuclid algorithmEuclid domainEuler characteristicEuler circuitEuler functionEuler graphEuler numberEuler pathEuler polyhedron formula Euler tourEuler traileven permutationeventeverywhere defined excess capacity existence proof existential generalization existential quantification existential quantifier existential specification explicitextended Fibonacci number extended Lucas number extensionextension field extension graphexterior algebraF facefactorfactorablefactotialfactorizationfaithful (full) functor Ferrers graphFibonacci numberfieldfilterfinite dimensional associative division algebra finite extensionfinite field (Galois field )finite groupfinite setfinitely generated modulefirst order theory with equalityfive-color theoremfive-time-repetitionfixed pointfloor functionflowforestforgetful functorfour-color theorem (conjecture)F-reduced productfree elementfree monoidfree occurrencefree R-modulefree variablefree-Ω-algebrafull n-treefunction schemeG Galileo paradoxGauss coefficientGBN (G?del-Bernays-von Neumann system) GCD (Greatest Common Divisor) generalized Petersen graphgenerating functiongenerating proceduregeneratorgenerator matrixgeneric elementgenusgirthG?del completeness theoremgolden section numbergraceful graphgraceful tree conjecturegraphgraph of first class for edge coloring graph of second class for edge coloring graph rankgraph sequencegreatest common factorgreatest elementgreedy algorithmGrelling paradoxGr?tzsch graphgroupgroup codegroup of graphgrowth of functionHajós conjectureHamilton cycleHamilton graphHamilton pathHarary graphhash functionHasse diagramHeawood graphheightHerschel graphhom functorhomemorphism homomorphism homomorphism image homomorphism of graph hyperoctahedronhypothelical syllogism hypothesis (premise)idealidempotentidentityidentity functionidentity natural transformation imageimbeddingimmediate predcessor immediate successorimpossible eventincidentincident axiomincident matrixinclusion and exclusion principle inclusion relationindegreeindependentindependent number independent setindependent transcendental element indexindirected method H Iindividual variableinduced subgraphinfinite extensioninfinite groupinfinite setinitial endpointinitial objectinjectioninjection functorinjective (one to one mapping) inner faceinner neighbour setinorder searchintegral domainintegral subdomaininternal direct sum intersectionintersection of graph intersection operation intervalinvariant factorinvariant factor idealinverseinverse limitinverse morphisminverse natural transformation inverse operationinverse relationinversioninvertableinvolution property irreflexiveisolated vertexisomorphic categoryisomorphismisomorphism of graphjoinjoin of graphJ Jordan algebraJordan product (anti-commutator)Jordan sieve formulaj-skewjuxtapositionk-chromatic graphk-connected graphk-critical graphk-edge chromatic graphk-edge-connected graphk-edge-critical graph Kanaugh mapkernelKirkman schoolgirl problem Klein 4 groupKonisberge Brudge problem Kruskal's algorithm Kuratowski theoremlabeled graphLah numberLatin rectangleLatin squarelatticelattice homomorphismlawLCM (Least Common Multiple) leader cosetleast elementleafleast upper boundleft (right) identityleft (right) invertible element left (right) moduleleft (right) zeroleft (right) zero divisorleft adjoint functorleft cancellableleft cosetlengthlexicographic orderlLie algebraline- grouplinear array (list)linear graphlinear order (total order)K Llinear order set (chain)logical connective logical followlogically equivanlent logically implies logically valid loopLucas numbermagicmany valued proposition logic map coloring problem matchingmathematical structure matrix representation maximal element maximal idealmaximal outerplanar graph maximal planar graph maximum flow maximum matching maxtermmaxterm normal form (conjunctive normal form)McGee graph meetMenger theorem Meredith graph message word mini term minimal -connected graph minimal polynomial minimal spanning tree Minimanoff paradox minimum distance Minkowski summinterm (fundamental conjunctive form)minterm normal form (disjunctive normal form)M?bius function M?bius ladder M?bius transform (inversion)modal logic modelmodule homomorphismMkmoduler latticemodulusmodus ponensmodus tollensmodule isomorphismmonic morphismmonoidmonomorphismmorphism (arrow)M?bius functionM?bius ladderM?bius transform (inversion)multigraphmultinomial coefficientmultinomial expansion theoremmultiple-error-correcting codemultiplication principlemutually exclusivemultiplication tablemutually orthogonal Latin squareN n-ary operationn-ary productn-ary tree (n-tree)n-tuplenatural deduction systemnatural homomorphismnatural isomorphismnatural transformationnearest neighbernegationneighbour setnext state transition functionnon-associative algebranon-standard logicNorlund formulanormal formnormal modelnormal subgroup (invariant subgroup)n-relationnull graph (discrete graph)null objectnullary operationobjectodd permutationoffspringone to oneone-to-one correspondence (bijection) onto optimal solutionorbitorderorder (lower order,same order) order ideal order relationordered pairOre conditionorientationorthogonal Latin square orthogonal layoutoutarcoutdegreeouter faceouter neighbourouterneighbour setouterplanar graphpancycle graphparallelismparallelism classparentparity-check codeparity-check equationparity-check machineparity-check matrixpartial functionpartial ordering (partial relation) partial order relation partial order set (poset)partitionpartition number of integerpartition number of setPascal formulapathperfect code O Pperfect t-error-correcting code perfect graph permutationpermutation grouppermutation with repetation Petersen graphp-graphPierce arrowpigeonhole principleplanar graphplane graphPolish formPólya theorempolynomailpolynomial codepolynomial representation polynomial ring positional treepossible worldpostorder searchpower functorpower of graphpower setpredicateprenex normal formpreorder searchpre-ordered setprimary cycle modulePRIM's algorithmprimeprime fieldprime to each otherprimitive connectiveprimitive elementprimitive polynomialprincipal idealprincipal ideal domainprinciple of dualityprinciple of mathematical induction principle of redundancy probabilisticprobability (theory)productproduct categoryproduct partial orderproduct-sum formproof (deduction)proof by contraditionproper coloringproper factorproper filterproper subgroupproperly inclusive relationproposition (statement)propositional constantpropositional formula (well-formed formula,wff) propositional functionpropositional variablepseudocodepullbackpushoutquantification theoryquantifierquasi order relationquaternionquotient (difference) algebraquotient algebraquotient field (field of fraction)quotient groupquotient modulequotient ring (difference ring , residue ring) quotient set Ramsey graph Ramsey number Ramsey theorem rangerankreachability reconstruction conjecture recursive redundant digits reflexiveregular expression regular graph R Qregular representationrelation matrixrelative setremainderreplacement theoremrepresentationrepresentation functorrestricted proposition formrestrictionretractionreverse Polish formRichard paradoxright adjoint functorright cancellableright factorright zero divisonringring of endomorphismring with unity elementR-linear independencerooted treeroot fieldrule of inferenceRussell paradoxS sample spacesatisfiablesaturatedscopesearchingsectionself-complement graphsemantical completenesssemantical consistentsemigroupseparable elementseparable extensionsequencesequentsequentialSheffer strokesiblingssimple algebraic extensionsimple cyclesimple extensionsimple graphsimple pathsimple proposition (atomic proposition) simple transcental extension simplicationsinkslopesmall categorysmallest element Socrates argument soundness (validity) theorem sourcespanning subgraph spanning treespectra of graphspetral radiussplitting fieldsquare matrixstandard modelstandard monomil statement (proposition) Steiner tripleStirling numberStirling transformstrong induction subalgebrasubcategorysubdirect product subdivison of graph subfieldsubformulasubdivision of graph subgraphsubgroupsub-modulesubmonoidsublatticesubrelationsubringsub-semigroup subscript。
混合循环图的特征值
混合循环图的特征值作者：许英来源：《教育教学论坛》 2014年第14期许英（新疆财经大学应用数学学院，新疆乌鲁木齐830012）摘要：一个图的邻接矩阵的特征值我们称为这个图的特征值，在物理和化学领域中，通过对物质分子所对应的分子图的特征值的研究，可以预知该物质在某些物理和化学方面的性质。
而在计算机网络中，研究网络对应的图的特征值将为深入研究该网络提供一个非常有用的代数工具。
因此，计算特殊图类的特征值是图谱理论中令大家感兴趣的问题。
在这篇文章中，我们研究了混合循环图和混合循环有向图的特征值的问题。
关键词：混合循环图；邻接矩阵；特征值中图分类号：G642.3 文献标志码：A 文章编号：1674-9324（2014）14-0128-02设G是一个单位元为1的有限群，S是G\1的一个子集。
群G关于集合S的Cayley有向图D=D（G，S）是一个点集为G的有向图，对于点g1，g2∈G，从g1到g2有一条弧当且仅当g2g1-1∈S。
如果S是逆闭的，即S=S-1，则Cayley有向图D（G，S）被认为是一个无向图，被称为群G关于S的Cayley图，表示为C（G，S）。
在文献[5]中，L.Lovasz确定了关于传递自同构群的谱。
在文献[1]中，L.Babai得到了关于群G不可约特征的Cayley图X（祝，S）的谱的表达式。
为了研究半对称图（正则边传递但不是点传递的图），文献[6]中定义了双Cayley 图。
设G是一个有限群，S是G的一个子集，双-Cayley图BC（G，S）是一个点集为G×{0，1}的二部图，边集为{{（g，0），（sg，1）}：g∈G，s∈S}。
当G是一个循环群时，双-Cayley图BC（G，S）被称为双循环图。
双-Cayley图可以推广到双-Cayley有向图上。
对于一个有限群G和群G的子集T1，T2，群G的关于T1和T2的双-Cayley有向图D=（V（D）），E（D）=D （G，T1，T2）被定义为二部有向图，点集为V（D）=G×{0，1}，并且对于点g1，g2∈G，（（g1 ，0），（g2 ，1））∈E（D）当且仅当g2=t1g1，其中t1∈T1；（（g1 ，1），（g2 ，0））∈E（D）当且仅当g1=t2g2，其中t2∈T2。
Yb∶Ca3(NbGa)5O12晶体的坩埚下降法生长及光学性能研究
第53卷第4期2024年4月人㊀工㊀晶㊀体㊀学㊀报JOURNAL OF SYNTHETIC CRYSTALS Vol.53㊀No.4April,2024YbʒCa 3(NbGa )5O 12晶体的坩埚下降法生长及光学性能研究赵㊀涛,艾㊀蕾,梁团结,钱慧宇,孙志刚,潘建国(宁波大学材料科学与化学工程学院,浙江省光电探测材料及器件重点实验室,宁波㊀315211)摘要:使用坩埚下降法成功生长出了镱离子掺杂钙铌镓石榴石晶体(YbʒCa 3(NbGa)5O 12)㊂通过XRD 测试分析了晶体的结构,该晶体为立方晶系,晶胞参数a =b =c =12.471Å㊂对该晶体进行了拉曼光谱㊁透过光谱㊁吸收和发射光谱㊁荧光寿命等测试,计算了该晶体的吸收截面㊁发射截面㊁增益截面等㊂研究了在空气中退火对该晶体吸收光谱㊁发射光谱㊁荧光寿命的影响,退火前在935nm 处吸收截面为1.82ˑ10-20cm 2,退火后降低为1.40ˑ10-20cm 2,退火前在1031nm 处的发射截面为0.56ˑ10-20cm 2,退火后降低为0.40ˑ10-20cm 2,退火前荧光衰减时间为1.42ms,退火后为1.32ms㊂结果表明,YbʒCa 3(NbGa)5O 12单晶在空气中退火会对晶体的激光性能造成不利影响㊂关键词:YbʒCa 3(NbGa)5O 12晶体;坩埚下降法;吸收光谱;发射光谱;荧光衰减;退火中图分类号:O782㊀㊀文献标志码:A ㊀㊀文章编号:1000-985X (2024)04-0620-07Growth and Optical Properties of YbʒCa 3(NbGa )5O 12Crystals by Bridgman MethodZHAO Tao ,AI Lei ,LIANG Tuanjie ,QIAN Huiyu ,SUN Zhigang ,PAN Jianguo(Key Laboratory of Photoelectric Detection Materials and Devices of Zhejiang Province,School of Materials Science and Chemical Engineering,Ningbo University,Ningbo 315211,China)Abstract :Ytterbium ion doped calcium niobium gallium garnet crystal (Yb ʒCa 3(NbGa)5O 12)was successfully grown by Bridgman method.The structure of the crystal was analyzed by XRD.The crystal is cubic crystal system,and the unit cell parameter a =b =c =12.471Å.The crystal was tested by Raman spectroscopy,transmission spectroscopy,absorption and emission spectroscopy,and fluorescence lifetime.The absorption cross section,emission cross section,and gain cross section of the crystal were calculated.The effects of annealing in air on the absorption spectrum,emission spectrum and fluorescence lifetime of the crystal were studied.The absorption cross section at 935nm before annealing is 1.82ˑ10-20cm 2,and it decreases to 1.40ˑ10-20cm 2after annealing.The emission cross section at 1031nm before annealing is 0.56ˑ10-20cm 2,and it decreases to 0.40ˑ10-20cm 2after annealing.The fluorescence decay time before annealing is 1.42ms,and it is 1.32ms after annealing.The results demonstrate that the annealing of YbʒCa 3(NbGa)5O 12single crystal in air will adversely affect the laser performance of the crystal.Key words :YbʒCa 3(NbGa)5O 12crystal;Bridgman method;absorption spectrum;emission spectrum;fluorescence decay;annealing㊀㊀㊀收稿日期:2023-12-08㊀㊀基金项目:国家自然科学基金(51832009,512302300)㊀㊀作者简介:赵㊀涛(1997 ),男,山西省人,硕士研究生㊂E-mail:1254983331@ ㊀㊀通信作者:孙志刚,博士,助理研究员㊂E-mail:sunzhigang@0㊀引㊀㊀言钙铌镓石榴石(CNGG)晶体是一类无序激光晶体,结构介于激光玻璃的无序结构和激光晶体的有序结构之间㊂无序结构的激光玻璃,是一类典型的非均匀加宽的激光增益介质,但玻璃具有长程无序结构,限制㊀第4期赵㊀涛等:YbʒCa3(NbGa)5O12晶体的坩埚下降法生长及光学性能研究621㊀了声子的平均自由程,导致其热学性能相对较差,限制了高效㊁高功率密度激光的获得[1]㊂而传统的激光晶体如钇铝石榴石(YAG)晶体,具有很好的热学性质,但长程有序的特点使其具有相对单一的激活离子取代位置,导致其配位单一,激活离子的光谱较窄[2]㊂无序的钙铌镓石榴石晶体兼具两者的优点,具有光谱的非均匀加宽特性和较高的热导率,使得其在激光领域中具有潜在的应用价值㊂NdʒCNGG晶体的具有较宽的吸收与发射光谱,Pan等[3]采用直拉法生长了无序的NdʒCNGG晶体,InGaAs LD泵浦的峰值吸收截面约为4.1ˑ10-20cm2,在808nm LD激发的发射荧光谱中,4F3/2ң4I11/2的半峰全宽(full width at half maximum, FWHM)为15nm,4F3/2ң4I13/2半峰全宽为27nm,在超快激光脉冲产生方面展示出巨大的潜力㊂目前,研究人员对NdʒCNGG晶体的连续波㊁调Q及锁模超短脉冲激光特性已做了大量㊁系统的研究[4-6]㊂20世纪90年代初,随着体积小㊁效率高㊁寿命长的LD泵浦源的出现,Yb3+作为激光基质激活离子的研究迅猛发展㊂Yb3+具有最简单的能级结构,与Nd3+相比,具有本征量子缺陷低,辐射量子效率高,能级寿命长,吸收和发射光谱宽等特点㊂特别是Yb3+的吸收峰位于900~1000nm,能与目前商用的InGaAs半导体激光二极管泵浦源有效耦合,并且不需要严格控制温度㊂YbʒCa3(NbGa)5O12晶体(YbʒCNGG)已有相关报道,可获得连续激光输出,并通过锁模和调Q获得脉冲激光输出[7-9],证明了YbʒCNGG在激光领域的潜在价值㊂目前报道的YbʒCNGG晶体都是使用提拉法生长,该晶体的坩埚下降法生长还没有报道㊂坩埚下降法生长晶体是在密闭环境中进行,能有效防止原料Ga2O3的挥发;此外,与提拉炉相比较,坩埚下降炉价格低廉,设备维护简单,使用坩埚下降法生长晶体能够极大地降低生产成本,因此YbʒCNGG晶体可能更适合使用坩埚下降法生长㊂本文成功使用坩埚下降法生长出较大尺寸的YbʒCNGG晶体,并开展了其光学性能研究㊂1㊀实㊀㊀验1.1㊀原料制备和晶体生长YbʒCNGG晶体在1450ħ左右一致熔融,但在高温下Ga2O3原料会挥发,因此本实验采用坩埚下降法,在密闭环境中生长该晶体㊂使用的原料为Yb2O3(纯99.99%),CaCO3(纯99.99%),Nb2O5(纯99.99%), Ga2O3(纯99.999%),采用Ca3Nb1.6875Ga3.1875O12成分配比,按照以下的化学反应式进行多晶料的合成㊂2.892CaCO3+0.813375Nb2O5+1.626375Ga2O3+0.054Yb2O3=0.964Ca3Nb1.6875Ga3.1875O12㊃0.036Yb3Ga5O12+2.892CO2(1)按上述配比称量原料,进行充分研磨,放入混料机中混合24h,再进行液压机压块,随后放入马弗炉进行第一次烧结,烧结温度1000ħ,保温10h;取出后再次研磨㊁压块,进行第二次烧结,烧结温度1250ħ,保温时间30h,得到YbʒCNGG的多晶料㊂将多晶料放进装有YAG[111]籽晶的铂金坩埚,放入坩埚下降炉中进行晶体生长㊂接种温度为1450ħ,下降速度8mm/d㊂晶体生长结束后,以20ħ/h左右的速率使炉温降至室温,以消除晶体生长过程中所产生的热应力㊂众所周知,激光晶体在高温环境中工作一段时间后,性能会有所降低㊂在高温㊁富氧或贫氧环境中工作一段时间后某些单晶会改变颜色,导致其光学吸收带发生变化,这种现象已经在硅酸铋[10]㊁铌酸盐[11-12]㊁磷酸盐[13]和碱金属钼酸盐[14-16]等氧化物中发现㊂因此,本文在空气中对YbʒCNGG晶体进行了热退火,以此来探究高温环境工作后晶体的光学性能变化㊂将加工好的一块晶片切成两块,其中一块放进马弗炉中,在空气氛围下进行退火,退火温度为1000ħ,保温时间10h㊂1.2㊀性能测试使用德国Bruker XRD D8Advance型X射线粉末衍射仪对YbʒCNGG晶体的粉末样品进行XRD测试,辐射源为Cu靶X射线管,工作电压和电流分别为40kV和40mA,扫描范围10ʎ~70ʎ,步幅为0.02ʎ㊂使用DXR3Raman Microscope光谱仪记录了晶体在295K下的拉曼光谱,激发源为532nm波长的激光㊂使用美国Lambda950型紫外可见近红外分光光度计测量了晶体的吸收和透过光谱㊂使用法国FL3-111型荧光光谱仪测试了晶体的发射光谱,激发源为980nm激光㊂采用英国FLS980荧光光谱仪测试了晶体的荧光衰减曲线,激发波长980nm,监测波长1031nm㊂622㊀研究论文人工晶体学报㊀㊀㊀㊀㊀㊀第53卷2㊀结果与讨论2.1㊀晶体生长图1(a)为采用坩埚下降法生长得到的YbʒCNGG晶体,晶体直径为25mm,接种后生长部分长度约为80mm,其中偏析层部分约为25mm㊂晶体呈现咖啡色,透明,内部有少量裂纹,晶体开裂与晶体自身性质以及生长工艺有关㊂图1(b)为加工后的YbʒCNGG晶片,晶片直径25mm,厚度为1mm,属于(111)晶面,晶片中横向裂纹是加工所致㊂图1㊀坩埚下降法生长的YbʒCNGG晶体Fig.1㊀YbʒCNGG crystals grown by Bridgman method2.2㊀XRD分析图2为YbʒCNGG晶体单晶部分和顶部偏析层部分的粉末XRD图谱,将单晶部分的XRD数据导入Jade 中,通过拟合得出该晶体是Ia3d空间群,属于立方晶系,晶胞参数a=b=c=12.471Å,α=β=γ=90ʎ,比已报道的CNGG晶体晶胞参数(12.51Å)略小[17],原因是掺杂的Yb3+半径小于被取代的Ca2+半径,导致晶体晶格收缩㊂通过Jade分析,顶部偏析层的杂质成分大部分是立方焦火成岩(Ca2Nb2O7),这与文献[18]中得出结论一致,原因是掺入Yb3+后,生成了镱镓石榴石(Yb3Ga5O12),导致Ca2+与Nb5+的过量,从而生成了不属于石榴石相的Ca2Nb2O7㊂2.3㊀拉曼光谱图3是室温下YbʒCNGG退火前后晶体样品的拉曼图谱对比,孤立金属氧四面体基团[MO4](M代表Ga 和Nb)在700~900cm-1存在对称伸缩振动,这些[MO4]基团是石榴石晶格的结构单元,M阳离子进入到石榴石结构的d位[19]㊂在700~900cm-1看到两个密集的振动峰C1和C3,分别是[GaO4]和[NbO4]基团群的对称伸缩振动造成的,C1和C2峰下降明显,C3和C4变化较小的可能原因是晶体中部分Ga3+挥发,改变了晶体的结构和振动特性,影响了振动模式的活性㊂Ga3+挥发会对晶体中[GaO4]基团的对称伸缩振动产生影响㊂通常情况下,Ga O键连接可能会中断或减弱,这种情况可能导致对称伸缩振动变弱,在拉曼光谱中可能会表现为C1和C2峰强度下降㊂C2和C4分别是C1和C3的伴峰,此处出现峰,则代表[GaO4]和[NbO4]附近出现阳离子空位,峰强度越高,则代表阳离子空位浓度越高㊂从图中可以看出,退火后C2和C4处都出现了微弱的伴峰,表明在退火后的晶体中,阳离子空位浓度增加了,主要原因是高温退火后晶体表面的Ga3+浓度降低,但是幅度较小[20]㊂2.4㊀透过和吸收光谱退火前后晶体样品的透过图谱如图4(a)所示,600~2500nm的整体透过率接近80%,说明晶体质量较高,退火后晶体颜色变化不明显㊂图4(b)是YbʒCNGG晶体的吸收截面图,吸收峰对应Yb3+的2F7/2(基态)ң2F5/2(激发态)跃迁㊂基态2F7/2和激发态2F5/2分别被晶体场劈裂为4个和3个Stark能级,从基态多重态的几个Stark能级到激发态多重态2F7/2(0㊁1㊁2㊁3)ң2F5/2(0ᶄ㊁1ᶄ㊁2ᶄ)的电子跃迁大多数是声子辅助的,从而产生了相当宽的谱带㊂晶体退火前在935nm处吸收截面为1.82ˑ10-20cm2,退火后为1.40ˑ10-20cm2;退火前在971nm处吸收截面为1.22ˑ10-20cm2,退火后为1.03ˑ10-20cm2,退火后吸收截面明显降低㊂此外,㊀第4期赵㊀涛等:YbʒCa 3(NbGa)5O 12晶体的坩埚下降法生长及光学性能研究623㊀从图4(c)和4(d)可以计算得出,晶体退火前在935nm 处FWHM 为47.46nm,退火后为44.60nm;退火前在971nm 处FWHM 为23.47nm,退火后为23.86nm㊂退火后在935nm 处的FWHM 比退火前小了2.86nm㊂图2㊀YbʒCNGG 晶体中部单晶部分及顶部偏析层部分的粉末XRD 图谱Fig.2㊀Powder XRD patterns of the middle single crystal part and the top segregation layer of YbʒCNGGcrystal 图3㊀室温下退火前后YbʒCNGG 晶体样品的拉曼图谱Fig.3㊀Raman spectra of YbʒCNGG crystal samples before and post annealing at roomtemperature图4㊀室温下退火前后YbʒCNGG 晶体样品的性能测试㊂(a)透过光谱;(b)吸收光谱;(c)退火前晶体样品吸收光谱的高斯拟合图;(d)退火后晶体样品吸收光谱的高斯拟合图Fig.4㊀Performance testing of YbʒCNGG crystal samples before and post annealing at room temperature.(a)Transmission spectra;(b)absorption spectra;(c)Gaussian fitting of absorption spectra of crystal sample before annealing;(d)Gaussian fitting of the absorption spectrum of crystal sample post annealing 2.5㊀发射光谱关于YbʒCNGG 晶体的发射截面σem (λ)计算,本文使用互易法(reciprocity method),用下列公式进行计算㊂624㊀研究论文人工晶体学报㊀㊀㊀㊀㊀㊀第53卷σem (λ)=σαbsZ l Z u exp E zl -hc λkT ()(2)式中:σabs 为吸收截面,h 为普朗克常数,k 为玻耳兹曼常数,c 为光速,λ为波长,T 为实验温度,Z l /Z u 为下㊁上能级的配分函数比,E zl 为零声子线㊂如图5(a)所示,计算得出退火前975nm 处的发射截面为1.28ˑ10-20cm 2,退火后为1.11ˑ10-20cm 2,退火前1031nm 处的发射截面为0.56ˑ10-20cm 2,退火后为0.40ˑ10-20cm 2㊂退火后975㊁1031nm 处的发射截面均低于退火前㊂图5(b)是在980nm 激光激发下得到的发射光谱,发射峰位于1031nm 处,在相同测试条件下,退火后该晶体的发射强度明显低于退火前,这与计算得出的结果相一致,表明YbʒCNGG 晶体在空气中退火后,对其激光性能有不利影响㊂原因是空气中的高温退火可能会对材料的物理和化学性质产生影响,包括晶格结构的变化和缺陷的生成㊂退火过程中晶格结构的变化和缺陷的形成可能对透过谱和发射谱性能产生影响㊂晶格结构变化:高温退火可能引起晶格结构的重新排列㊂在退火过程中,原子或分子在晶体中重新定位以达到更低的能量状态㊂这可能导致晶格略微变化,晶格参数可能发生微小的变化,如晶胞参数㊁晶体取向等㊂这种微小的结构变化可能会影响透过谱和发射谱的特性㊂缺陷的生成:高温退火也可能导致缺陷的生成㊂例如,点缺陷(Ga 3+的挥发)㊁位错或晶界等缺陷的产生㊂这些缺陷可能导致电子状态的变化㊁局部晶格畸变或者在晶体中引入能级㊂这些缺陷可能会影响材料的光学性质,包括透过谱和发射谱㊂图5㊀室温下退火前后YbʒCNGG 晶体样品的发射截面曲线(a)和980nm 激光激发下得到的发射光谱(b)Fig.5㊀Emission cross-section curves (a)and emission spectra at 980nm excitation (b)of YbʒCNGG crystal samples before and post annealing at room temperature2.6㊀增益截面根据上述吸收和发射截面光谱,增益截面σg (λ)可由下式计算:σg (λ)=βσem (λ)-(1-β)σabs (λ)(3)式中:β为激发态离子反转分数㊂图6所示为退火前后的YbʒCNGG 晶体样品在不同β值(0,0.25,0.50,0.75,1.00)下的增益截面曲线㊂如图6(a)所示,在1010~1040nm 处,当布居反转分数达到25%时,增益截面变为正值㊂如此低的反转比例意味着1031nm 波长的YbʒCNGG 激光器将具有较低的泵浦阈值,这表明YbʒCNGG 晶体是1031nm 激光器的理想候选材料㊂在高抽运情况下,增益截面谱也较宽,表现出良好的可协调性㊂而退火后该晶体增益截面曲线如图6(b)所示,并且在布居反转比例达到50%时,在1031nm 附近的增益带宽明显低于退火前,因此理论上通过被动锁模达到最小脉冲也将会受到影响[21],也就是说,在高温下工作会对该晶体超快激光的产生造成不利影响㊂2.7㊀荧光衰减室温下对退火前后的YbʒCNGG 晶体样品进行荧光衰减测试㊂如图7所示,激发波长980nm,监测波长1031nm,采用单指数函数拟合,如公式(4)所示㊂y =A 1e -x t +y 0(4)㊀第4期赵㊀涛等:YbʒCa3(NbGa)5O12晶体的坩埚下降法生长及光学性能研究625㊀式中:A1为前因子,y0为初始强度,t为时间,x㊁y为测试的横纵坐标,对应波长㊁强度㊂通过拟合得到退火前的荧光衰减时间为1.42ms,退火后的荧光衰减时间为1.32ms,观察到退火后Yb3+的寿命减少,表明这种退火在晶体中引入了进一步的缺陷,很可能是由表面Ga3+的挥发造成的,与文献中采用提拉法生长的YbʒCNGG晶体τ=816μs相比较,结果相差很大,可能是该晶体有很强的重吸收,造成直接测量荧光寿命不准确,但是与文献中退火后Yb3+的寿命会减少的结论是一致的[20]㊂图6㊀室温下退火前后YbʒCNGG晶体样品增益截面曲线Fig.6㊀YbʒCNGG crystal samples gain cross-section curves before and post annealing at room temperature图7㊀室温下退火前后YbʒCNGG晶体样品荧光衰减曲线Fig.7㊀YbʒCNGG crystal samples fluorescence decay curves before and post annealing at room temperature3㊀结㊀㊀论采用坩埚下降法,生长出尺寸为ϕ25mmˑ80mm的YbʒCNGG透明单晶,通过XRD粉末衍射,得出了偏析层的主要杂质成分为Ca2Nb2O7㊂通过透过和吸收光谱得出该晶体退火前在935和971nm处有很宽的吸收带宽,分别为47.46和23.47nm,退火后935nm处吸收带宽变窄㊂尽管常规情况下退火有助于提高晶体的均匀性和激光性能,但在本文中通过对YbʒCNGG晶体退火前后晶体发射截面和增益截面的计算,以及发射光谱和荧光衰减的测量,发现采用高温退火可能会引入缺陷并导致激光性能下降㊂这可能暗示着退火温度需要重新评估或者退火周期需要调整以更好地维持晶体性能,后续本团队会继续研究不同退火条件对YbʒCNGG晶体激光性能的影响㊂参考文献[1]㊀于浩海,潘忠奔,张怀金,等.无序激光晶体及其超快激光研究进展[J].人工晶体学报,2021,50(4):648-668+583.YU H H,PAN Z B,ZHANG H J,et al.Development of disordered laser crystals and their ultrafast lasers[J].Journal of Synthetic Crystals, 2021,50(4):648-668+583(in Chinese).[2]㊀KANCHANAVALEERAT E,COCHET-MUCHY D,KOKTA M,et al.Crystal growth of high doped NdʒYAG[J].Optical Materials,2004,26626㊀研究论文人工晶体学报㊀㊀㊀㊀㊀㊀第53卷(4):337-341.[3]㊀PAN H,PAN Z B,CHU H W,et al.GaAs Q-switched NdʒCNGG lasers:operating at4F3/2ң2I11/2and4F3/2ң2I13/2transitions[J].OpticsExpress,2019,27(11):15426-15432.[4]㊀SHI Z B,FANG X,ZHANG H J,et al.Continuous-wave laser operation at1.33μm of NdʒCNGG and NdʒCLNGG crystals[J].Laser 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A phase transition for the diameter of the configuration model Remco van der Hofstad
A phase transition for the diameter of the configuration modelRemco van der Hofstad∗Gerard Hooghiemstra†and Dmitri Znamenski‡August31,2007AbstractIn this paper,we study the configuration model(CM)with i.i.d.degrees.We establisha phase transition for the diameter when the power-law exponentτof the degrees satisfiesτ∈(2,3).Indeed,we show that forτ>2and when vertices with degree2are present withpositive probability,the diameter of the random graph is,with high probability,bounded frombelow by a constant times the logarithm of the size of the graph.On the other hand,assumingthat all degrees are at least3or more,we show that,forτ∈(2,3),the diameter of the graphis,with high probability,bounded from above by a constant times the log log of the size of thegraph.1IntroductionRandom graph models for complex networks have received a tremendous amount of attention in the past decade.See[1,22,26]for reviews on complex networks and[2]for a more expository account.Measurements have shown that many real networks share two fundamental properties. Thefirst is the fact that typical distances between vertices are small,which is called the‘small world’phenomenon(see[27]).For example,in the Internet,IP-packets cannot use more than a threshold of physical links,and if the distances in terms of the physical links would be large,e-mail service would simply break down.Thus,the graph of the Internet has evolved in such a way that typical distances are relatively small,even though the Internet is rather large.The second and maybe more surprising property of many networks is that the number of vertices with degree k falls offas an inverse power of k.This is called a‘power law degree sequence’,and resulting graphs often go under the name‘scale-free graphs’(see[15]for a discussion where power laws occur in the Internet).The observation that many real networks have the above two properties has incited a burst of activity in network modelling using random graphs.These models can,roughly speaking,be divided into two distinct classes of models:‘static’models and’dynamic’models.In static models, we model with a graph of a given size a snap-shot of a real network.A typical example of this kind of model is the configuration model(CM)which we describe below.A related static model, which can be seen as an inhomogeneous version of the Erd˝o s-R´e nyi random graph,is treated in great generality in[4].Typical examples of the‘dynamical’models,are the so-called preferential attachment models(PAM’s),where added vertices and edges are more likely to be attached to vertices that already have large degrees.PAM’s often focus on the growth of the network as a way to explain the power law degree sequences.∗Department of Mathematics and Computer Science,Eindhoven University of Technology,P.O.Box513,5600 MB Eindhoven,The Netherlands.E-mail:rhofstad@win.tue.nl†Delft University of Technology,Electrical Engineering,Mathematics and Computer Science,P.O.Box5031,2600 GA Delft,The Netherlands.E-mail:G.Hooghiemstra@ewi.tudelft.nl‡EURANDOM,P.O.Box513,5600MB Eindhoven,The Netherlands.E-mail:znamenski@eurandom.nlPhysicists have predicted that distances in PAM’s behave similarly to distances in the CM with similar degrees.Distances in the CM have attracted considerable attention(see e.g.,[14,16,17,18]), but distances in PAM’s far less(see[5,19]),which makes it hard to verify this prediction.Together with[19],the current paper takes afirst step towards a rigorous verification of this conjecture.At the end of this introduction we will return to this observation,but let usfirst introduce the CM and present our diameter results.1.1The configuration modelThe CM is defined as follows.Fix an integer N.Consider an i.i.d.sequence of random variables D1,D2,...,DN.We will construct an undirected graph with N vertices where vertex j has degreeD j.We will assume that LN =Nj=1D j is even.If LNis odd,then we will increase DNby1.Thissingle change will make hardly any difference in what follows,and we will ignore this effect.We will later specify the distribution of D1.To construct the graph,we have N separate vertices and incident to vertex j,we have D j stubs or half-edges.The stubs need to be paired to construct the graph.We number the stubs in a givenorder from1to LN .We start by pairing at random thefirst stub with one of the LN−1remainingstubs.Once paired,two stubs form a single edge of the graph.Hence,a stub can be seen as the left-or the right-half of an edge.We continue the procedure of randomly choosing and pairing the stubs until all stubs are connected.Unfortunately,vertices having self-loops,as well as multiple edges between vertices,may occur,so that the CM is a multigraph.However,self-loops are scarce when N→∞,as shown e.g.in[7].The above model is a variant of the configuration model[3],which,given a degree sequence,is the random graph with that given degree sequence.The degree sequence of a graph is the vector of which the k th coordinate equals the fraction of vertices with degree k.In our model,by the law of large numbers,the degree sequence is close to the distribution of the nodal degree D of which D1,...,DNare i.i.d.copies.The probability mass function and the distribution function of the nodal degree law are denoted byP(D=k)=f k,k=1,2,...,and F(x)= xk=1f k,(1.1)where x is the largest integer smaller than or equal to x.We pay special attention to distributions of the form1−F(x)=x1−τL(x),(1.2) whereτ>2and L is slowly varying at infinity.This means that the random variables D j obey a power law,and the factor L is meant to generalize the model.We denote the expectation of D byµ,i.e.,µ=∞k=1kf k.(1.3)1.2The diameter in the configuration modelIn this section we present the results on the bounds on the diameter.We use the abbreviation whp for a statement that occurs with probability tending to1if the number of vertices of the graph N tends to∞.Theorem1.1(Lower bound on diameter)Forτ>2,assuming that f1+f2>0and f1<1, there exists a positive constantαsuch that whp the diameter of the configuration model is bounded below byαlog N.A more precise result on the diameter in the CM is presented in[16],where it is proved that under rather general assumptions on the degree sequence of the CM,the diameter of the CM divided by log N converges to a constant.This result is also valid for related models,such as the Erd˝o s-R´e nyi random graph,but the proof is quite difficult.While Theorem1.1is substantially weaker,the fact that a positive constant times log N appears is most interesting,as we will discuss now in more detail.Indeed,the result in Theorem1.1is most interesting in the case whenτ∈(2,3).By[18, Theorem1.2],the typical distance forτ∈(2,3)is proportional to log log N,whereas we show here that the diameter is bounded below by a positive constant times log N when f1+f2>0and f1<1. Therefore,we see that the average distance and the diameter are of a different order of magnitude. The pairs of vertices where the distance is of the order log N are thus scarce.The proof of Theorem 1.1reveals that these pairs are along long lines of vertices with degree2that are connected to each other.Also in the proof of[16],one of the main difficulties is the identification of the precise length of these long thin lines.Our second main result states that whenτ∈(2,3),the above assumption that f1+f2>0is necessary and sufficient for log N lower bounds on the diameter.In Theorem1.2below,we assume that there exists aτ∈(2,3)such that,for some c>0and all x≥1,1−F(x)≥cx1−τ,(1.4) which is slightly weaker than the assumption in(1.2).We further define for integer m≥2and a real numberσ>1,CF =CF(σ,m)=2|log(τ−2)|+2σlog m.(1.5)Then our main upper bound on the diameter when(1.4)holds is as follows:Theorem1.2(A log log upper bound on the diameter)Fix m≥2,and assume that P(D≥m+1)=1,and that(1.4)holds.Then,for everyσ>(3−τ)−1,the diameter of the configuration model is,whp,bounded above by CFlog log N.1.3Discussion and related workTheorem1.2has a counterpart for preferential attachment models(PAM)proved in[19].In these PAM’s,at each integer time t,a new vertex with m≥1edges attached to it,is added to the graph. The new edges added at time t are then preferentially connected to older edges,i.e.,conditionally on the graph at time t−1,which is denoted by G(t−1),the probability that a given edge is connected to vertex i is proportional to d i(t−1)+δ,whereδ>−m is afixed parameter and d i(t−1)is the degree of vertex i at time t−1.A substantial literature exists,see e.g.[10],proving that the degree sequence of PAM’s in rather great generality satisfy a power law(see e.g.the references in [11]).In the above setting of linear preferential attachment,the exponentτis equal to[21,11]τ=3+δm.(1.6)A log log t upper bound on the diameter holds for PAM’s with m≥2and−m<δ<0,which, by(1.6),corresponds toτ∈(2,3)[19]:Theorem1.3(A log log upper bound on the diameter of the PAM)Fix m≥2andδ∈(−m,0).Then,for everyσ>13−τ,and withCG (σ)=4|log(τ−2)|+4σlog mthe diameter of the preferential attachment model is,with high probability,bounded above by CGlog log t,as t→∞.Observe that the condition m≥2in the PAM corresponds to the condition P(D≥m+1)=1 in the CM,where one half-edge is used to attach the vertex,while in PAM’s,vertices along a pathhave degree at least three when m≥2.Also note from the definition of CG and CFthat distancesin PAM’s tend to be twice as big compared to distances in the CM.This is related to the structure of the graphs.Indeed,in both graphs,vertices of high degree play a crucial role in shortest paths. In the CM vertices of high degree are often directly connected to each other,while in the PAM, they tend to be connected through a later vertex which links to both vertices of high degree.Unfortunately,there is no log t lower bound in the PAM forδ>0and m≥2,or equivalently τ>3.However,[19]does contain a(1−ε)log t/log log t lower bound for the diameter when m≥1 andδ≥0.When m=1,results exists on log t asymptotics of the diameter,see e.g.[6,24].The results in Theorems1.1–1.3are consistent with the non-rigorous physics predictions that distances in the PAM and in the CM,for similar degree sequences,behave similarly.It is an interesting problem,for both the CM and PAM,to determine the exact constant C≥0such that the diameter of the graph of N vertices divided by log N converges in probability to C.For the CM,the results in[16]imply that C>0,for the PAM,this is not known.We now turn to related work.Many distance results for the CM are known.Forτ∈(1,2) distances are bounded[14],forτ∈(2,3),they behave as log log N[25,18,9],whereas forτ>3 the correct scaling is log N[17].Observe that these results induce lower bounds for the diameter of the CM,since the diameter is the supremum of the distance,where the supremum is taken over all pairs of vertices.Similar results for models with conditionally independent edges exist,see e.g. [4,8,13,23].Thus,for these classes of models,distances are quite well understood.The authors in [16]prove that the diameter of a sparse random graph,with specified degree sequence,has,whp, diameter equal to c log N(1+o(1)),for some constant c.Note that our Theorems1.1–1.2imply that c>0when f1+f2>0,while c=0when f1+f2=0and(1.4)holds for someτ∈(2,3).There are few results on distances or diameter in PAM’s.In[5],it was proved that in the PAMand forδ=0,for whichτ=3,the diameter of the resulting graph is equal to log tlog log t (1+o(1)).Unfortunately,the matching result for the CM has not been proved,so that this does not allow us to verify whether the models have similar distances.This paper is organized as follows.In Section2,we prove the lower bound on the diameter formulated in Theorem1.1and in Section3we prove the upper bound in Theorem1.2.2A lower bound on the diameter:Proof of Theorem1.1We start by proving the claim when f2>0.The idea behind the proof is simple.Under the conditions of the theorem,one can,whp,find a pathΓ(N)in the random graph such that this path consists exclusively of vertices with degree2and has length at least2αlog N.This implies that the diameter is at leastαlog N,since the above path could be a cycle.Below we define a procedure which proves the existence of such a path.Consider the process of pairing stubs in the graph.We are free to choose the order in which we pair the free stubs,since this order is irrelevant for the distribution of the random graph.Hence,we are allowed to start with pairing the stubs of the vertices of degree2.Let N(2)be the number of vertices of degree2and SN(2)=(i1,...,i N(2))∈N N(2)the collection of these vertices.We will pair the stubs and at the same time define a permutationΠ(N)=(i∗1, (i)N(2))of SN(2),and a characteristicχ(N)=(χ1,...,χN(2))onΠ(N),whereχj is either0or1.Π(N)andχ(N)will be defined inductively in such a way that for any vertex i∗k ∈Π(N),χk=1,if and only if vertex i∗k is connected to vertex i∗k+1.Hence,χ(N)contains a substring ofat least2αlog N ones precisely when the random graph contains a pathΓ(N)of length at least 2αlog N.We initialize our inductive definition by i∗1=i1.The vertex i∗1has two stubs,we consider the second one and pair it to an arbitrary free stub.If this free stub belongs to another vertex j=i∗1 in SN(2)then we choose i∗2=j andχ1=1,otherwise we choose i∗2=i2,andχ1=0.Suppose forsome1<k≤N(2),the sequences(i∗1, (i)k )and(χ1,...,χk−1)are defined.Ifχk−1=1,then onestub of i∗k is paired to a stub of i∗k−1,and another stub of i∗kis free,else,ifχk−1=0,vertex i∗khastwo free stubs.Thus,for every k≥1,the vertex i∗k has at least one free stub.We pair this stubto an arbitrary remaining free stub.If this second stub belongs to vertex j∈SN (2)\{i∗1, (i)k},then we choose i∗k+1=j andχk=1,else we choose i∗k+1as thefirst stub in SN(2)\{i∗1, (i)k},andχk=0.Hence,we have defined thatχk=1precisely when vertex i∗k is connected to vertexi∗k+1.We show that whp there exists a substring of ones of length at least2αlog N in thefirsthalf ofχN ,i.e.,inχ12(N)=(χi∗1,...,χi∗N(2)/2).For this purpose,we couple the sequenceχ12(N)with a sequence B12(N)={ξk},whereξk are i.i.d.Bernoulli random variables taking value1withprobability f2/(4µ),and such that,whp,χi∗k ≥ξk for all k∈{1,..., N(2)/2 }.We write PNforthe law of the CM conditionally on the degrees D1,...,DN.Then,for any1≤k≤ N(2)/2 ,thePN-probability thatχk=1is at least2N(2)−CN(k)LN −CN(k),(2.1)where,as before,N(2)is the total number of vertices with degree2,and CN(k)is one plus thetotal number of paired stubs after k−1pairings.By definition of CN(k),for any k≤N(2)/2,we haveCN(k)=2(k−1)+1≤N(2).(2.2) Due to the law of large numbers we also have that whpN(2)≥f2N/2,LN≤2µN.(2.3) Substitution of(2.2)and(2.3)into(2.1)then yields that the right side of(2.1)is at leastN(2) LN ≥f24µ.Thus,whp,we can stochastically dominate all coordinates of the random sequenceχ12(N)with ani.i.d.Bernoulli sequence B12(N)of Nf2/2independent trials with success probability f2/(4µ)>0.It is well known(see e.g.[12])that the probability of existence of a run of2αlog N ones convergesto one whenever2αlog N≤log(Nf2/2) |log(f2/(4µ))|,for some0< <1.We conclude that whp the sequence B1(N)contains a substring of2αlog N ones.Since whpχN ≥B12(N),where the ordering is componentwise,whp the sequenceχNalso contains the samesubstring of2αlog N ones,and hence there exists a required path consisting of at least2αlog N vertices with degree2.Thus,whp the diameter is at leastαlog N,and we have proved Theorem 1.1in the case that f2>0.We now complete the proof of Theorem1.1when f2=0by adapting the above argument. When f2=0,and since f1+f2>0,we must have that f1>0.Let k∗>2be the smallest integer such that f k∗>0.This k∗must exist,since f1<1.Denote by N∗(2)the total number of vertices of degree k∗of which itsfirst k∗−2stubs are connected to a vertex with degree1.Thus,effectively, after thefirst k∗−2stubs have been connected to vertices with degree1,we are left with a structure which has2free stubs.These vertices will replace the N(2)vertices used in the above proof.It is not hard to see that whp N∗(2)≥f∗2N/2for some f∗2>0.Then,the argument for f2>0can be repeated,replacing N(2)by N∗(2)and f2by f∗2.In more detail,for any1≤k≤ N∗(2)/(2k∗) , the PN-probability thatχk=1is at least2N∗(2)−C∗N(k)LN −CN(k),(2.4)where C∗N(k)is the total number of paired stubs after k−1pairings of the free stubs incident to the N∗(2)vertices.By definition of C∗N(k),for any k≤N∗(2)/(2k∗),we haveCN(k)=2k∗(k−1)+1≤N∗(2).(2.5)Substitution of(2.5),N∗(2)≥f∗2N/2and the bound on LNin(2.3)into(2.4)gives us that the right side of(2.4)is at leastN∗(2) LN ≥f∗24µ.Now the proof of Theorem1.1in the case where f2=0and f1∈(0,1)can be completed as above.We omit further details. 3A log log upper bound on the diameter forτ∈(2,3)In this section,we investigate the diameter of the CM when P(D≥m+1)=1,for some integer m≥2.We assume(1.4)for someτ∈(2,3).We will show that under these assumptions CFlog log N isan upper bound on the diameter of the CM,where CFis defined in(1.5).The proof is divided into two key steps.In thefirst,in Proposition3.1,we give a bound on the diameter of the core of the CM consisting of all vertices with degree at least a certain power of log N.This argument is very close in spirit to the one in[25],the only difference being that we have simplified the argument slightly.After this,in Proposition3.4,we derive a bound on the distance between vertices with small degree and the core.We note that Proposition3.1only relies on the assumption in(1.4),while Proposition3.4only relies on the fact that P(D≥m+1)=1,for some m≥2.The proof of Proposition3.1can easily be adapted to a setting where the degrees arefixed, by formulating the appropriate assumptions on the number of vertices with degree at least x for a sufficient range of x.This assumption would replace(1.5).Proposition3.4can easily be adapted to a setting where there are no vertices of degree smaller than or equal to m.This assumption would replace the assumption P(D≥m+1)=1,for some m≥2.We refrain from stating these extensions of our results,and start by investigating the core of the CM.We takeσ>13−τand define the core CoreNof the CM to beCoreN={i:D i≥(log N)σ},(3.1)i.e.,the set of vertices with degree at least(log N)σ.Also,for a subset A⊆{1,...,N},we define the diameter of A to be equal to the maximal shortest path distance between any pair of vertices of A.Note,in particular,that if there are pairs of vertices in A that are not connected,then the diameter of A is infinite.Then,the diameter of the core is bounded in the following proposition:Proposition3.1(Diameter of the core)For everyσ>13−τ,the diameter of CoreNis,whp,bounded above by2log log N|log(τ−2)|(1+o(1)).(3.2)Proof.We note that(1.4)implies that whp the largest degree D(N)=max1≤i≤N D i satisfiesD(N)≥u1,where u1=N1τ−1(log N)−1,(3.3) because,when N→∞,P(D(N)>u1)=1−P(D(N)≤u1)=1−(F(u1))N≥1−(1−cu1−τ1)N=1−1−c(log N)τ−1NN∼1−exp(−c(log N)τ−1)→1.(3.4)DefineN (1)={i :D i ≥u 1},(3.5)so that,whp ,N (1)=∅.For some constant C >0,which will be specified later,and k ≥2we define recursively u k =C log N u k −1 τ−2,and N (k )={i :D i ≥u k }.(3.6)We start by identifying u k :Lemma 3.2(Identification of u k )For each k ∈N ,u k =C a k (log N )b k N c k ,(3.7)with c k =(τ−2)k −1τ−1,b k =13−τ−4−τ3−τ(τ−2)k −1,a k =1−(τ−2)k −13−τ.(3.8)Proof .We will identify a k ,b k and c k recursively.We note that,by (3.3),c 1=1τ−1,b 1=−1,a 1=0.By (3.6),we can,for k ≥2,relate a k ,b k ,c k to a k −1,b k −1,c k −1as follows:c k =(τ−2)c k −1,b k =1+(τ−2)b k −1,a k =1+(τ−2)a k −1.(3.9)As a result,we obtain c k =(τ−2)k −1c 1=(τ−2)k −1τ−1,(3.10)b k =b 1(τ−2)k −1+k −2 i =0(τ−2)i =1−(τ−2)k −13−τ−(τ−2)k −1,(3.11)a k =1−(τ−2)k −13−τ.(3.12)The key step in the proof of Proposition 3.1is the following lemma:Lemma 3.3(Connectivity between N (k −1)and N (k ))Fix k ≥2,and C >4µ/c (see (1.3),and (1.4)respectively).Then,the probability that there exists an i ∈N (k )that is not directly connected to N (k −1)is o (N −γ),for some γ>0independent of k .Proof .We note that,by definition,i ∈N (k −1)D i ≥u k −1|N (k −1)|.(3.13)Also,|N (k −1)|∼Bin N,1−F (u k −1) ,(3.14)and we have that,by (1.4),N [1−F (u k −1)]≥cN (u k −1)1−τ,(3.15)which,by Lemma 3.2,grows as a positive power of N ,since c k ≤c 2=τ−2τ−1<1τ−1.We use a concentration of probability resultP (|X −E [X ]|>t )≤2e −t 22(E [X ]+t/3),(3.16)which holds for binomial random variables[20],and gives that that the probability that|N(k−1)|is bounded below by N[1−F(u k−1)]/2is exponentially small in N.As a result,we obtain that for every k,and whpi∈N(k)D i≥c2N(u k)2−τ.(3.17)We note(see e.g.,[18,(4.34)]that for any two sets of vertices A,B,we have thatPN (A not directly connected to B)≤e−D A D BL N,(3.18)where,for any A⊆{1,...,N},we writeD A=i∈AD i.(3.19)On the event where|N(k−1)|≥N[1−F(u k−1)]/2and where LN≤2µN,we then obtain by(3.18),and Boole’s inequality that the PN-probability that there exists an i∈N(k)such that i is not directly connected to N(k−1)is bounded byNe−u k Nu k−1[1−F(u k−1)]2L N≤Ne−cu k(u k−1)2−τ4µ=N1−cC4µ,(3.20)where we have used(3.6).Taking C>4µ/c proves the claim. We now complete the proof of Proposition3.1.Fixk∗= 2log log N|log(τ−2)|.(3.21)As a result of Lemma3.3,we have whp that the diameter of N(k∗)is at most2k∗,because thedistance between any vertex in N(k∗)and the vertex with degree D(N)is at most k∗.Therefore,weare done when we can show thatCoreN⊆N(k∗).(3.22)For this,we note thatN(k∗)={i:D i≥u k∗},(3.23)so that it suffices to prove that u k∗≥(log N)σ,for anyσ>13−τ.According to Lemma3.2,u k∗=C a k∗(log N)b k∗N c k∗.(3.24) Because for x→∞,and2<τ<3,x(τ−2)2log x|log(τ−2)|=x·x−2=o(log x),(3.25) wefind with x=log N thatlog N·(τ−2)2log log N|log(τ−2)|=o(log log N),(3.26) implying that N c k∗=(log N)o(1),(log N)b k∗=(log N)13−τ+o(1),and C a k∗=(log N)o(1).Thus,u k∗=(log N)13−τ+o(1),(3.27)so that,by picking N sufficiently large,we can make13−τ+o(1)≤σ.This completes the proof ofProposition3.1. For an integer m≥2,we defineC(m)=σ/log m.(3.28)Proposition 3.4(Maximal distance between periphery and core)Assume that P (D ≥m +1)=1,for some m ≥2.Then,for every σ>(3−τ)−1the maximal distance between any vertex and the core is,whp ,bounded from above by C (m )log log N .Proof .We start from a vertex i and will show that the probability that the distance between i and Core N is at least C (m )log log N is o (N −1).This proves the claim.For this,we explore the neighborhood of i as follows.From i ,we connect the first m +1stubs (ignoring the other ones).Then,successively,we connect the first m stubs from the closest vertex to i that we have connected to and have not yet been explored.We call the arising process when we have explored up to distance k from the initial vertex i the k -exploration tree .When we never connect two stubs between vertices we have connected to,then the number of vertices we can reach in k steps is precisely equal to (m +1)m k −1.We call an event where a stub on the k -exploration tree connects to a stub incident to a vertex in the k -exploration tree a collision .The number of collisions in the k -exploration tree is the number of cycles or self-loops in it.When k increases,the probability of a collision increases.However,for k of order log log N ,the probability that more than two collisions occur in the k -exploration tree is small,as we will prove now:Lemma 3.5(Not more than one collision)Take k = C (m )log log N .Then,the P N -probab-ility that there exists a vertex of which the k -exploration tree has at least two collisions,before hittingthe core Core N ,is bounded by (log N )d L −2N ,for d =4C (m )log (m +1)+2σ.Proof .For any stub in the k -exploration tree,the probability that it will create a collision beforehitting the core is bounded above by (m +1)m k −1(log N )σL −1N .The probability that two stubs will both create a collision is,by similar arguments,bounded above by (m +1)m k −1(log N )σL −1N2.The total number of possible pairs of stubs in the k -exploration tree is bounded by[(m +1)(1+m +...+m k −1)]2≤[(m +1)m k ]2,so that,by Boole’s inequality,the probability that the k -exploration tree has at least two collisions is bounded by (m +1)m k 4(log N )2σL −2N .(3.29)When k = C (m )log log N ,we have that (m +1)m k 4(log N )2σ≤(log N )d ,where d is defined inthe statement of the lemma. Finally,we show that,for k = C (m )log log N ,the k -exploration tree will,whp connect to the Core N :Lemma 3.6(Connecting exploration tree to core)Take k = C (m )log log N .Then,the probability that there exists an i such that the distance of i to the core is at least k is o (N −1).Proof .Since µ<∞we have that L N /N ∼µ.Then,by Lemma 3.5,the probability that there exists a vertex for which the k -exploration tree has at least 2collisions before hitting the core is o (N −1).When the k -exploration tree from a vertex i does not have two collisions,then there are at least (m −1)m k −1stubs in the k th layer that have not yet been connected.When k = C (m )log log N this number is at least equal to (log N )C (m )log m +o (1).Furthermore,the expected number of stubs incident to the Core N is at least N (log N )σP (D 1≥(log N )σ)so that whp the number of stubs incident to Core N is at least (compare (1.4))12N (log N )σP (D 1≥(log N )σ)≥c 2N (log N )2−τ3−τ.(3.30)By(3.18),the probability that we connect none of the stubs in the k th layer of the k-exploration tree to one of the stubs incident to CoreNis bounded byexp−cN(log N)2−τ3−τ+C(m)log m2LN≤exp−c4µ(log N)2−τ3−τ+σ=o(N−1),(3.31)because whp LN /N≤2µ,and since2−τ3−τ+σ>1.Propositions3.1and3.4prove that whp the diameter of the configuration model is boundedabove by CF log log N,with CFdefined in(1.5).This completes the proof of Theorem1.2.Acknowledgements.The work of RvdH and DZ was supported in part by Netherlands Organ-isation for Scientific Research(NWO).References[1]R.Albert and A.-L.Barab´a si.Statistical mechanics of complex networks.Rev.Mod.Phys.74,47-97,(2002).[2]A.-L.Barab´a si.Linked,The New Science of Networks.Perseus Publishing,Cambridge,Mas-sachusetts,(2002).[3]B.Bollob´a s.Random Graphs,2nd edition,Academic Press,(2001).[4]B.Bollob´a s,S.Janson,and O.Riordan.The phase transition in inhomogeneous randomgraphs.Random Structures and Algorithms31,3-122,(2007).[5]B.Bollob´a s and O.Riordan.The diameter of a scale-free random binatorica,24(1):5–34,(2004).[6]B.Bollob´a s and O.Riordan.Shortest paths and load scaling in scale-free trees.Phys.Rev.E.,69:036114,(2004).[7]T.Britton,M.Deijfen,and A.Martin-L¨o f.Generating simple random graphs with prescribeddegree distribution.J.Stat.Phys.,124(6):1377–1397,(2006).[8]F.Chung and L.Lu.The average distances in random graphs with given expected degrees.A,99(25):15879–15882(electronic),(2002).[9]R.Cohen and S.Havlin.Scale free networks are ultrasmall,Physical Review Letters90,058701,(2003).[10]C.Cooper and A.Frieze.A general model of web graphs.Random Structures Algorithms,22(3):311–335,(2003).[11]M.Deijfen,H.van den Esker,R.van der Hofstad and G.Hooghiemstra.A preferential attach-ment model with random initial degrees.Preprint(2007).To appear in Arkiv f¨o r Matematik.[12]P.Erd¨o s and A.R´e nyi.On a new law of large numbers,J.Analyse Math.23,103–111,(1970).[13]H.van den Esker,R.van der Hofstad and G.Hooghiemstra.Universality forthe distance infinite variance random graphs.Preprint(2006).Available from http://ssor.twi.tudelft.nl/∼gerardh/[14]H.van den Esker,R.van der Hofstad,G.Hooghiemstra and D.Znamenski.Distances inrandom graphs with infinite mean degrees,Extremes8,111-141,2006.。
非手性液晶聚合物中超分子手性结构的诱导与光照固定
Vol. 36 No. 1功 能 高 分 子 学 报2023 年 2 月Journal of Functional Polymers21文章编号： 1008-9357(2023)01-0021-10DOI: 10.14133/ki.1008-9357.20220723001非手性液晶聚合物中超分子手性结构的诱导与光照固定郭玉权1， 贺子翔1， 程笑笑1， 张 弓1， 缪腾飞1,2， 张 伟1（1. 苏州大学材料与化学化工学部, 新型功能高分子材料国家地方联合工程实验室, 江苏省新型功能高分子材料工程实验室, 苏州市大分子设计与精密合成重点实验室, 江苏 苏州 215123；2. 淮阴师范学院化学化工学院, 江苏省低维材料化学重点实验室, 江苏 淮安 223300）摘 要： 超分子手性结构由于其形成所需的非共价相互作用力较弱，因此极易在外界条件刺激下发生解离，从而导致手性消失。
在绝大多数情况下，以非手性单元为组装模块的组装体系中超分子手性结构在外部环境刺激下被破坏后无法恢复。
为了提高超分子手性结构的稳定性，克服其对原有手性源的依赖性，设计合成了一种侧链型非手性苯甲酸苯酯液晶聚合物，在聚合物侧链末端引入肉桂酸结构作为交联基团。
将手性柠檬烯的手性诱导与简单的光照交联结合即可实现超分子手性的共价键固定，极大地简化了交联过程，为手性材料的制备与手性信息存储提供了新的策略。
关键词： 超分子手性；液晶聚合物；手性诱导；光照交联；手性存储；光照固定中图分类号： O631.2 文献标志码： AInduction and Photo-Locking of Supramolecular Chiral Structures inAchiral Liquid Crystalline PolymersGUO Yuquan1, HE Zixiang1, CHENG Xiaoxiao1, ZHANG Gong1, MIAO Tengfei1,2, ZHANG Wei1（1. State and Local Joint Engineering Laboratory for Novel Functional Polymeric Materials, Jiangsu Engineering Laboratory of Novel Functional Polymeric Materials, Suzhou Key Laboratory of Macromolecular Design and Precision Synthesis, College of Chemistry, Chemical Engineering and Materials Science, Soochow University, Suzhou 215123, Jiangsu, China;2. Jiangsu Key Laboratory for Chemistry of Low-Dimensional Materials, School of Chemistry and Chemical Engineering,Huaiyin Normal University, Huaian 223300, Jiangsu, China）Abstract:Supramolecular chirality usually plays an important role in biological processes and living systems. A novel achiral liquid crystalline polymer where the molecular chirality can be transferred is described, which provides a simple method to realize the chiral selfassembly in achiral polymer systems. However, the supramolecular chiral structures are often easily destroyed by external stimulus due to the weak non-covalent interactions between building blocks, which limits the application of the chiral supramolecular materials to a large extent. In the most systems composed of achiral precursors, the destroyed chiral ordered structures cannot be recovered in the absence of the pre-chiral resource. In order to improve the收稿日期： 2022-07-23基金项目： 国家自然科学基金（21971180，92056111）；江苏省高等学校自然科学研究重大项目（19 KJA360006）作者简介： 郭玉权（1997—），男，硕士生，主要研究方向为超分子手性的调控与固定。
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nected, the pair 0 1 (n) =_!_n log n, 0 11(n) = n is a. pair of strong threshold functions for the property and the corresponding sharp-threshold distribution function is e-r-2ll'; thus we have proved1 that putting 1 N(n) = - n log n + y n+ o(n) we have
Cayley图论文：Cayley图正规Cayley图1-正则图图正则表示
Cayley图论文：Cayley图正规Cayley图 1-正则图图正则表示【中文摘要】设G是有限群,S是G的不包含单位元1的生成子集,|S|=3.群G的一个Cayley图X= Cay(G,S)称为正规的,如果右乘变换群R(G)在AutX中正规.本文中,我们对除交换群、二面体群、四元数群、拟二面体群、半二面体群之外的所有8p阶群的3度Cayley图进行了分类,并在此基础上得到一类新的3度1-正则Cayley图.【英文摘要】Let G be a finite group and S a generating subsetof G.|S|= 3. A Cayley graph X= Cay(G.S) is said to be normalif R(G) is normal in AutX. In this paper, we classify 3-valent Cayley graphs of order 8p except those of abelian groups, dihedral groups, quaternion groups, quasi-dihedral groups and semi-dihedral groups, and as a result, we obtain a new infinite families of 3-valent one-regular Cayley graphs.【关键词】Cayley图正规Cayley图 1-正则图图正则表示【英文关键词】Cayley graph normal Cayley graph one-regular graph DRR【目录】8p阶3度Cayley图摘要4-5Abstract5引言7-9一预备知识9-21§1.1 基本概念和命题9-10§1.2 群Gi(i=1,2, (6)的一些性质10-21二主要结果及证明21-37§2.1 定理1的证明21§2.2 定理2的证明21§2.3 定理3的证明21-28§2.4 定理4的证明28-32§2.5 定理5的证明32-34§2.6 定理6的证明34-37参考文献37-39致谢39【索购全文找】1.3.9.9.3.8.8.4.81.3.8.1.1.3.7.2.1同时提供论文写作一对一辅导和论文发表服务。
Statistical mechanics of complex networks,RevModPhys.74.47
Statistical mechanics of complex networksRe´ka Albert*and Albert-La´szlo´Baraba´siDepartment of Physics,University of Notre Dame,Notre Dame,Indiana46556(Published30January2002)Complex networks describe a wide range of systems in nature and society.Frequently cited examples include the cell,a network of chemicals linked by chemical reactions,and the Internet,a network of routers and computers connected by physical links.While traditionally these systems have been modeled as random graphs,it is increasingly recognized that the topology and evolution of real networks are governed by robust organizing principles.This article reviews the recent advances in the field of complex networks,focusing on the statistical mechanics of network topology and dynamics.After reviewing the empirical data that motivated the recent interest in networks,the authors discuss the main models and analytical tools,covering random graphs,small-world and scale-free networks, the emerging theory of evolving networks,and the interplay between topology and the network’s robustness against failures and attacks.CONTENTSI.Introduction48II.The Topology of Real Networks:Empirical Results49A.World Wide Web49B.Internet50C.Movie actor collaboration network52D.Science collaboration graph52E.The web of human sexual contacts52F.Cellular networks52G.Ecological networks53H.Phone call network53I.Citation networks53works in linguistics53 K.Power and neural networks54 L.Protein folding54 III.Random-Graph Theory54A.The Erdo˝s-Re´nyi model54B.Subgraphs55C.Graph evolution56D.Degree distribution57E.Connectedness and diameter58F.Clustering coefficient58G.Graph spectra59 IV.Percolation Theory59A.Quantities of interest in percolation theory60B.General results601.The subcritical phase(pϽp c)602.The supercritical phase(pϾp c)61C.Exact solutions:Percolation on a Cayley tree61D.Scaling in the critical region62E.Cluster structure62F.Infinite-dimensional percolation62G.Parallels between random-graph theory andpercolation63 V.Generalized Random Graphs63A.Thresholds in a scale-free random graph64B.Generating function formalism64ponent sizes and phase transitions652.Average path length65C.Random graphs with power-law degreedistribution66D.Bipartite graphs and the clustering coefficient66 VI.Small-World Networks67A.The Watts-Strogatz model67B.Properties of small-world networks681.Average path length682.Clustering coefficient693.Degree distribution704.Spectral properties70 VII.Scale-Free Networks71A.The Baraba´si-Albert model71B.Theoretical approaches71C.Limiting cases of the Baraba´si-Albert model73D.Properties of the Baraba´si-Albert model741.Average path length742.Node degree correlations753.Clustering coefficient754.Spectral properties75 VIII.The Theory of Evolving Networks76A.Preferential attachment⌸(k)761.Measuring⌸(k)for real networks762.Nonlinear preferential attachment773.Initial attractiveness77B.Growth781.Empirical results782.Analytical results78C.Local events791.Internal edges and rewiring792.Internal edges and edge removal79D.Growth constraints801.Aging and cost802.Gradual aging81petition in evolving networks811.Fitness model812.Edge inheritance82F.Alternative mechanisms for preferentialattachment821.Copying mechanism822.Edge redirection823.Walking on a network834.Attaching to edges83G.Connection to other problems in statisticalmechanics831.The Simon model832.Bose-Einstein condensation85*Present address:School of Mathematics,University of Min-nesota,Minneapolis,Minnesota55455.REVIEWS OF MODERN PHYSICS,VOLUME74,JANUARY20020034-6861/2002/74(1)/47(51)/$35.00©2002The American Physical Society47IX.Error and Attack Tolerance86A.Numerical results861.Random network,random node removal872.Scale-free network,random node removal873.Preferential node removal87B.Error tolerance:analytical results88C.Attack tolerance:Analytical results89D.The robustness of real networks90munication networks902.Cellular networks913.Ecological networks91X.Outlook91A.Dynamical processes on networks91B.Directed networks92C.Weighted networks,optimization,allometricscaling92D.Internet and World Wide Web93E.General questions93F.Conclusions94 Acknowledgments94 References94 I.INTRODUCTIONComplex weblike structures describe a wide variety of systems of high technological and intellectual impor-tance.For example,the cell is best described as a com-plex network of chemicals connected by chemical reac-tions;the Internet is a complex network of routers and computers linked by various physical or wireless links; fads and ideas spread on the social network,whose nodes are human beings and whose edges represent various social relationships;the World Wide Web is an enormous virtual network of Web pages connected by hyperlinks.These systems represent just a few of the many examples that have recently prompted the scien-tific community to investigate the mechanisms that de-termine the topology of complex networks.The desire to understand such interwoven systems has encountered significant challenges as well.Physics,a major benefi-ciary of reductionism,has developed an arsenal of suc-cessful tools for predicting the behavior of a system as a whole from the properties of its constituents.We now understand how magnetism emerges from the collective behavior of millions of spins,or how quantum particles lead to such spectacular phenomena as Bose-Einstein condensation or superfluidity.The success of these mod-eling efforts is based on the simplicity of the interactions between the elements:there is no ambiguity as to what interacts with what,and the interaction strength is uniquely determined by the physical distance.We are at a loss,however,to describe systems for which physical distance is irrelevant or for which there is ambiguity as to whether two components interact.While for many complex systems with nontrivial network topology such ambiguity is naturally present,in the past few years we have increasingly recognized that the tools of statistical mechanics offer an ideal framework for describing these interwoven systems as well.These developments have introduced new and challenging problems for statistical physics and unexpected links to major topics in condensed-matter physics,ranging from percolation to Bose-Einstein condensation.Traditionally the study of complex networks has been the territory of graph theory.While graph theory ini-tially focused on regular graphs,since the1950s large-scale networks with no apparent design principles have been described as random graphs,proposed as the sim-plest and most straightforward realization of a complex network.Random graphs werefirst studied by the Hun-garian mathematicians Paul Erdo˝s and Alfre´d Re´nyi. According to the Erdo˝s-Re´nyi model,we start with N nodes and connect every pair of nodes with probability p,creating a graph with approximately pN(NϪ1)/2 edges distributed randomly.This model has guided our thinking about complex networks for decades since its introduction.But the growing interest in complex sys-tems has prompted many scientists to reconsider this modeling paradigm and ask a simple question:are the real networks behind such diverse complex systems as the cell or the Internet fundamentally random?Our in-tuition clearly indicates that complex systems must dis-play some organizing principles,which should be at some level encoded in their topology.But if the topology of these networks indeed deviates from a random graph, we need to develop tools and measurements to capture in quantitative terms the underlying organizing prin-ciples.In the past few years we have witnessed dramatic ad-vances in this direction,prompted by several parallel de-velopments.First,the computerization of data acquisi-tion in allfields led to the emergence of large databases on the topology of various real networks.Second,the increased computing power allowed us to investigate networks containing millions of nodes,exploring ques-tions that could not be addressed before.Third,the slow but noticeable breakdown of boundaries between disci-plines offered researchers access to diverse databases, allowing them to uncover the generic properties of com-plex networks.Finally,there is an increasingly voiced need to move beyond reductionist approaches and try to understand the behavior of the system as a whole.Along this route,understanding the topology of the interac-tions between the components,i.e.,networks,is un-avoidable.Motivated by these converging developments and cir-cumstances,many new concepts and measures have been proposed and investigated in depth in the past few years.However,three concepts occupy a prominent place in contemporary thinking about complex net-works.Here we define and briefly discuss them,a discus-sion to be expanded in the coming sections.Small worlds:The small-world concept in simple terms describes the fact that despite their often large size,in most networks there is a relatively short path between any two nodes.The distance between two nodes is defined as the number of edges along the short-est path connecting them.The most popular manifesta-tion of small worlds is the‘‘six degrees of separation’’concept,uncovered by the social psychologist Stanley Milgram(1967),who concluded that there was a path of48R.Albert and A.-L.Baraba´si:Statistical mechanics of complex networks Rev.Mod.Phys.,Vol.74,No.1,January2002acquaintances with a typical length of about six betweenmost pairs of people in the United States(Kochen,1989).The small-world property appears to characterizemost complex networks:the actors in Hollywood are onaverage within three co-stars from each other,or thechemicals in a cell are typically separated by three reac-tions.The small-world concept,while intriguing,is notan indication of a particular organizing principle.In-deed,as Erdo˝s and Re´nyi have demonstrated,the typi-cal distance between any two nodes in a random graphscales as the logarithm of the number of nodes.Thusrandom graphs are small worlds as well.Clustering:A common property of social networks isthat cliques form,representing circles of friends or ac-quaintances in which every member knows every othermember.This inherent tendency to cluster is quantifiedby the clustering coefficient(Watts and Strogatz,1998),a concept that has its roots in sociology,appearing underthe name‘‘fraction of transitive triples’’(Wassermannand Faust,1994).Let us focusfirst on a selected node iin the network,having k i edges which connect it to k iother nodes.If the nearest neighbors of the originalnode were part of a clique,there would be k i(k iϪ1)/2 edges between them.The ratio between the number E iof edges that actually exist between these k i nodes andthe total number k i(k iϪ1)/2gives the value of the clus-tering coefficient of node i,C iϭ2E ik i͑k iϪ1͒.(1)The clustering coefficient of the whole network is the average of all individual C i’s.An alternative definition of C that is often used in the literature is discussed in Sec.VI.B.2(Barrat and Weigt,2000;Newman,Strogatz, and Watts,2000).In a random graph,since the edges are distributed randomly,the clustering coefficient is Cϭp(Sec.III.F). However,in most,if not all,real networks the clustering coefficient is typically much larger than it is in a compa-rable random network(i.e.,having the same number of nodes and edges as the real network).Degree distribution:Not all nodes in a network have the same number of edges(same node degree).The spread in the node degrees is characterized by a distri-bution function P(k),which gives the probability that a randomly selected node has exactly k edges.Since in a random graph the edges are placed randomly,the major-ity of nodes have approximately the same degree,close to the average degree͗k͘of the network.The degree distribution of a random graph is a Poisson distribution with a peak at P(͗k͘).One of the most interesting de-velopments in our understanding of complex networks was the discovery that for most large networks the de-gree distribution significantly deviates from a Poisson distribution.In particular,for a large number of net-works,including the World Wide Web(Albert,Jeong, and Baraba´si,1999),the Internet(Faloutsos et al.,1999), or metabolic networks(Jeong et al.,2000),the degree distribution has a power-law tail,P͑k͒ϳkϪ␥.(2) Such networks are called scale free(Baraba´si and Al-bert,1999).While some networks display an exponential tail,often the functional form of P(k)still deviates sig-nificantly from the Poisson distribution expected for a random graph.These discoveries have initiated a revival of network modeling in the past few years,resulting in the introduc-tion and study of three main classes of modeling para-digms.First,random graphs,which are variants of the Erdo˝s-Re´nyi model,are still widely used in manyfields and serve as a benchmark for many modeling and em-pirical studies.Second,motivated by clustering,a class of models,collectively called small-world models,has been proposed.These models interpolate between the highly clustered regular lattices and random graphs.Fi-nally,the discovery of the power-law degree distribution has led to the construction of various scale-free models that,by focusing on the network dynamics,aim to offer a universal theory of network evolution.The purpose of this article is to review each of these modeling efforts,focusing on the statistical mechanics of complex networks.Our main goal is to present the the-oretical developments in parallel with the empirical data that initiated and support the various models and theo-retical tools.To achieve this,we start with a brief de-scription of the real networks and databases that repre-sent the testing ground for most current modeling efforts.II.THE TOPOLOGY OF REAL NETWORKS:EMPIRICAL RESULTSThe study of most complex networks has been initi-ated by a desire to understand various real systems, ranging from communication networks to ecological webs.Thus the databases available for study span sev-eral disciplines.In this section we review briefly those that have been studied by researchers aiming to uncover the general features of complex networks.Beyond a de-scription of the databases,we shall focus on three robust measures of a network’s topology:average path length, clustering coefficient,and degree distribution.Other quantities,as discussed in the following sections,will again be tested on these databases.The properties of the investigated databases,as well as the obtained expo-nents,are summarized in Tables I and II.A.World Wide WebThe World Wide Web represents the largest network for which topological information is currently available. The nodes of the network are the documents(web pages)and the edges are the hyperlinks(URL’s)that point from one document to another(see Fig.1).The size of this network was close to one billion nodes at the end of1999(Lawrence and Giles,1998,1999).The in-terest in the World Wide Web as a network boomed after it was discovered that the degree distribution of the web pages follows a power law over several orders of magnitude(Albert,Jeong,and Baraba´si,1999;Kumar49R.Albert and A.-L.Baraba´si:Statistical mechanics of complex networks Rev.Mod.Phys.,Vol.74,No.1,January2002et al.,1999).Since the edges of the World Wide Web aredirected,the network is characterized by two degree dis-tributions:the distribution of outgoing edges,P out(k), signifies the probability that a document has k outgoinghyperlinks,and the distribution of incoming edges, P in(k),is the probability that k hyperlinks point to acertain document.Several studies have established that both P out(k)and P in(k)have power-law tails: P out͑k͒ϳkϪ␥out and P in͑k͒ϳkϪ␥in.(3) Albert,Jeong,and Baraba´si(1999)have studied asubset of the World Wide Web containing325729nodes and have found␥outϭ2.45and␥inϭ2.1.Kumar et al. (1999)used a40-million-document crawl by Alexa Inc., obtaining␥outϭ2.38and␥inϭ2.1(see also Kleinberg et al.,1999).A later survey of the World Wide Web to-pology by Broder et al.(2000)used two1999Altavistacrawls containing in total200million documents,obtain-ing␥outϭ2.72and␥inϭ2.1with scaling holding close to five orders of magnitude(Fig.2).Adamic and Huber-man(2000)used a somewhat different representation of the World Wide Web,with each node representing a separate domain name and two nodes being connected if any of the pages in one domain linked to any page in the other.While this method lumped together pages that were on the same domain,representing a nontrivial ag-gregation of the nodes,the distribution of incoming edges still followed a power law with␥in domϭ1.94.Note that␥in is the same for all measurements at the document level despite the two-years’time delay be-tween thefirst and last web crawl,during which the World Wide Web had grown at leastfive times larger. However,␥out has a tendency to increase with the sample size or time(see Table II).Despite the large number of nodes,the World Wide Web displays the small-world property.This wasfirst re-ported by Albert,Jeong,and Baraba´si(1999),who found that the average path length for a sample of 325729nodes was11.2and predicted,usingfinite size scaling,that for the full World Wide Web of800million nodes that would be a path length of around19.Subse-quent measurements by Broder et al.(2000)found that the average path length between nodes in a50-million-node sample of the World Wide Web is16,in agreement with thefinite size prediction for a sample of this size. Finally,the domain-level network displays an average path length of3.1(Adamic,1999).The directed nature of the World Wide Web does not allow us to measure the clustering coefficient using Eq.(1).One way to avoid this difficulty is to make the net-work undirected,making each edge bidirectional.This was the path followed by Adamic(1999),who studied the World Wide Web at the domain level using a1997 Alexa crawl of50million web pages distributed among 259794sites.Adamic removed the nodes that had have only one edge,focusing on a network of153127sites. While these modifications are expected to increase the clustering coefficient somewhat,she found Cϭ0.1078, orders of magnitude higher than C randϭ0.00023corre-sponding to a random graph of the same size and aver-age degree.B.InternetThe Internet is a network of physical links between computers and other telecommunication devices(Fig.TABLE I.The general characteristics of several real networks.For each network we have indicated the number of nodes,the average degree͗k͘,the average path length l,and the clustering coefficient C.For a comparison we have included the average path length l rand and clustering coefficient C rand of a random graph of the same size and average degree.The numbers in the last column are keyed to the symbols in Figs.8and9.Network Size͗k͘l l rand C C rand Reference Nr. WWW,site level,undir.15312735.21 3.1 3.350.10780.00023Adamic,19991 Internet,domain level3015–6209 3.52–4.11 3.7–3.76 6.36–6.180.18–0.30.001Yook et al.,2001a,Pastor-Satorras et al.,20012 Movie actors22522661 3.65 2.990.790.00027Watts and Strogatz,19983 LANL co-authorship529099.7 5.9 4.790.43 1.8ϫ10Ϫ4Newman,2001a,2001b,2001c4 MEDLINE co-authorship152025118.1 4.6 4.910.066 1.1ϫ10Ϫ5Newman,2001a,2001b,2001c5 SPIRES co-authorship56627173 4.0 2.120.7260.003Newman,2001a,2001b,2001c6 NCSTRL co-authorship11994 3.599.77.340.4963ϫ10Ϫ4Newman,2001a,2001b,2001c7 Math.co-authorship70975 3.99.58.20.59 5.4ϫ10Ϫ5Baraba´si et al.,20018 Neurosci.co-authorship20929311.56 5.010.76 5.5ϫ10Ϫ5Baraba´si et al.,20019 E.coli,substrate graph2827.35 2.9 3.040.320.026Wagner and Fell,200010 E.coli,reaction graph31528.3 2.62 1.980.590.09Wagner and Fell,200011 Ythan estuary food web1348.7 2.43 2.260.220.06Montoya and Sole´,200012 Silwood Park food web154 4.75 3.40 3.230.150.03Montoya and Sole´,200013 Words,co-occurrence460.90270.13 2.67 3.030.4370.0001Ferrer i Cancho and Sole´,200114 Words,synonyms2231113.48 4.5 3.840.70.0006Yook et al.,2001b15 Power grid4941 2.6718.712.40.080.005Watts and Strogatz,199816C.Elegans28214 2.65 2.250.280.05Watts and Strogatz,199817 50R.Albert and A.-L.Baraba´si:Statistical mechanics of complex networksRev.Mod.Phys.,Vol.74,No.1,January20021).The topology of the Internet is studied at two differ-ent levels.At the router level,the nodes are the routers,and edges are the physical connections between them.At the interdomain (or autonomous system)level,eachwork structure of the World Wide Web and the Internet.Upper panel:the nodes of the World Wide Web are web documents,connected with directed hyperlinks (URL’s).Lower panel:on the Internet the nodes are the routers and computers,and the edges are the wires and cables that physi-cally connect them.Figure courtesy of Istva´n Albert.TABLE II.The scaling exponents characterizing the degree distribution of several scale-free networks,for which P (k )follows apower law (2).We indicate the size of the network,its average degree ͗k ͘,and the cutoff for the power-law scaling.For directed networks we list separately the indegree (␥in )and outdegree (␥out )exponents,while for the undirected networks,marked with an asterisk (*),these values are identical.The columns l real ,l rand ,and l pow compare the average path lengths of real networks with power-law degree distribution and the predictions of random-graph theory (17)and of Newman,Strogatz,and Watts (2001)[also see Eq.(63)above],as discussed in Sec.V .The numbers in the last column are keyed to the symbols in Figs.8and workSize͗k ͘␥out ␥in lreallrandlpowReference Nr.WWW 325729 4.51900 2.45 2.111.28.32 4.77Albert,Jeong,and Baraba´si 19991WWW 4ϫ1077 2.38 2.1Kumar et al.,19992WWW 2ϫ1087.54000 2.72 2.1168.857.61Broder et al.,20003WWW,site 260000 1.94Huberman and Adamic,20004Internet,domain *3015–4389 3.42–3.7630–40 2.1–2.2 2.1–2.246.3 5.2Faloutsos,19995Internet,router *3888 2.5730 2.48 2.4812.158.757.67Faloutsos,19996Internet,router *150000 2.6660 2.4 2.41112.87.47Govindan,20007Movie actors *21225028.78900 2.3 2.3 4.54 3.65 4.01Baraba´si and Albert,19998Co-authors,SPIRES *566271731100 1.2 1.24 2.12 1.95Newman,2001b9Co-authors,neuro.*20929311.54400 2.1 2.16 5.01 3.86Baraba´si et al.,200110Co-authors,math.*70975 3.9120 2.5 2.59.58.2 6.53Baraba´si et al.,200111Sexual contacts *2810 3.4 3.4Liljeros et al.,200112Metabolic,E.coli 7787.4110 2.2 2.2 3.2 3.32 2.89Jeong et al.,200013Protein,S.cerev.*1870 2.39 2.4 2.4Jeong,Mason,et al.,200114Ythan estuary *1348.735 1.05 1.052.43 2.26 1.71Montoya and Sole´,200014Silwood Park *154 4.7527 1.13 1.13 3.43.232Montoya and Sole´,200016Citation 7833398.573Redner,199817Phone call 53ϫ1063.16 2.1 2.1Aiello et al.,200018Words,co-occurrence *46090270.13 2.7 2.7Ferrer i Cancho and Sole´,200119Words,synonyms *2231113.48 2.8 2.8Yook et al.,2001b20FIG.2.Degree distribution of the World Wide Web from twodifferent measurements:ᮀ,the 325729-node sample of Albert et al.(1999);᭺,the measurements of over 200million pages by Broder et al.(2000);(a)degree distribution of the outgoing edges;(b)degree distribution of the incoming edges.The data have been binned logarithmically to reduce noise.Courtesy of Altavista and Andrew Tomkins.The authors wish to thank Luis Amaral for correcting a mistake in a previous version of this figure (see Mossa et al.,2001).51R.Albert and A.-L.Baraba´si:Statistical mechanics of complex networks Rev.Mod.Phys.,Vol.74,No.1,January 2002domain,composed of hundreds of routers and comput-ers,is represented by a single node,and an edge is drawn between two domains if there is at least one route that connects them.Faloutsos et al.(1999)have studied the Internet at both levels,concluding that in each case the degree distribution follows a power law.The inter-domain topology of the Internet,captured at three dif-ferent dates between 1997and the end of 1998,resultedin degree exponents between ␥I as ϭ2.15and ␥I asϭ2.2.The 1995survey of Internet topology at the router level,containing 3888nodes,found ␥I rϭ2.48(Faloutsos et al.,1999).Recently Govindan and Tangmunarunkit (2000)mapped the connectivity of nearly 150000router inter-faces and nearly 200000router adjacencies,confirmingthe power-law scaling with ␥I rӍ2.3[see Fig.3(a)].The Internet as a network does display clustering and small path length as well.Yook et al.(2001a)and Pastor-Satorras et al.(2001),studying the Internet at the do-main level between 1997and 1999,found that its clus-tering coefficient ranged between 0.18and 0.3,to be compared with C rand Ӎ0.001for random networks with similar parameters.The average path length of the In-ternet at the domain level ranged between 3.70and 3.77(Pastor-Satorras et al.,2001;Yook et al.2001a)and at the router level it was around 9(Yook et al.,2001a),indicating its small-world character.C.Movie actor collaboration networkA much-studied database is the movie actor collabo-ration network,based on the Internet Movie Database,which contains all movies and their casts since the 1890s.In this network the nodes are the actors,and two nodes have a common edge if the corresponding actors have acted in a movie together.This is a continuously expand-ing network,with 225226nodes in 1998(Watts and Stro-gatz,1998),which grew to 449913nodes by May 2000(Newman,Strogatz,and Watts,2000).The average path length of the actor network is close to that of a random graph with the same size and average degree,3.65com-pared with 2.9,but its clustering coefficient is more than 100times higher than a random graph (Watts and Stro-gatz,1998).The degree distribution of the movie actor network has a power-law tail for large k [see Fig.3(b)],following P (k )ϳk Ϫ␥actor ,where ␥actor ϭ2.3Ϯ0.1(Bara-ba´si and Albert,1999;Albert and Baraba ´si,2000;Ama-ral et al.,2000).D.Science collaboration graphA collaboration network similar to that of the movie actors can be constructed for scientists,where the nodes are the scientists and two nodes are connected if the two scientists have written an article together.To uncover the topology of this complex graph,Newman (2001a,2001b,2001c)studied four databases spanning physics,biomedical research,high-energy physics,and computer science over a five-year window (1995–1999).All these networks show a small average path length but a high clustering coefficient,as summarized in Table I.The de-gree distribution of the collaboration network of high-energy physicists is an almost perfect power law with an exponent of 1.2[Fig.3(c)],while the other databases display power laws with a larger exponent in the tail.Baraba´si et al.(2001)investigated the collaboration graph of mathematicians and neuroscientists publishing between 1991and 1998.The average path length of these networks is around l math ϭ9.5and l nsci ϭ6,their clustering coefficient being C math ϭ0.59and C nsci ϭ0.76.The degree distributions of these collaboration networks are consistent with power laws with degree ex-ponents 2.1and 2.5,respectively [see Fig.3(d)].E.The web of human sexual contactsMany sexually transmitted diseases,including AIDS,spread on a network of sexual relationships.Liljeros et al.(2001)have studied the web constructed from the sexual relations of 2810individuals,based on an exten-sive survey conducted in Sweden in 1996.Since the edges in this network are relatively short lived,they ana-lyzed the distribution of partners over a single year,ob-taining for both females and males a power-law degree distribution with an exponent ␥f ϭ3.5Ϯ0.2and ␥m ϭ3.3Ϯ0.2,respectively.F .Cellular networksJeong et al.(2000)studied the metabolism of 43or-ganisms representing all three domains of life,recon-structing them in networks in which the nodes aretheFIG.3.The degree distribution of several real networks:(a)Internet at the router level.Data courtesy of Ramesh Govin-dan;(b)movie actor collaboration network.After Baraba´si and Albert 1999.Note that if TV series are included as well,which aggregate a large number of actors,an exponential cut-off emerges for large k (Amaral et al.,2000);(c)co-authorship network of high-energy physicists.After Newman (2001a,2001b);(d)co-authorship network of neuroscientists.AfterBaraba´si et al.(2001).52R.Albert and A.-L.Baraba´si:Statistical mechanics of complex networks Rev.Mod.Phys.,Vol.74,No.1,January 2002。
CurriculumVita-Chiao-Yao（Joe）She-CAS
Curriculum Vita - Chiao-Yao (Joe) SheEducation:1957 - B.S., Taiwan University, Taipei, Taiwan1961 - M.S., North Dakota State University, Fargo, North Dakota1964 - Ph.D., Stanford University, Stanford, CaliforniaExperience:1975-Present Professor of Physics, Colorado State University, Fort Collins, CO1968- 1971 Assistant Professor of Physics, Colorado State University1971- 1975 Associate Professor of Physics, Colorado State University1964 - 1968 Assistant Professor of Electrical Engineering, The University of Minnesota Honors, Memberships and Services:Fellow of the Optical Society of AmericaMember of APS, AGU1976 Research Publication Award, Naval Research Laboratory, Washington, D.C.1978 President of the Rocky Mountain Section of OSA.1987 Burlington Northern Faculty Achievement Award, Colorado State University 1988-1989 Golden Screw (Teaching) Award, Colorado State University1988, 1995 On NSF Review Panels for Research Initiation Awards, Lightwave Technology Program, and Optical Science and Engineering, respectively1992-1994 Member, AMS Committee on Laser Studies of the Atmosphere1993-1995 Member, CLEO/IQEC Program Committee1994-1997 Member, Arecibo Users and Scientific Advisory Committee, N.A.I.C.1989-2005 Fellow, Coop. Institute for Research in the Atmosphere, Colorado State University1997-2000 Member, NSF CEDAR Science Steering Committee2000-2001 Fulbright Research Award, Norway2003NSF/CEDAR Workshop – CEDAR Lecture Prize2003AGU Editor’s Citation – Outstanding Reviewer for Geophysics Research Letters 2005 Included in the 60th Diamond Edition of Marquis Who’s Who in AmericaBook Chapter and EditingWilliam B. Grant, Edward V. Browell, Robert T. Menzies, Kenneth Sassen and Chiao-Yao She (Editors), Selected Papers on Laser Applications in Remote Sensing, SPIE Milestone Series, MS 141 (1997).Research Interest and Current SupportThe research interests of Prof. She have been broad and often interdisciplinary. He has developed new laser measurement techniques for solving basic as well as applied problems. For the past 25 years, he has made a series of innovations in two high-spectral-resolution lidars: Rayleigh-Mie lidar and narrowband sodium lidar for atmospheric temperature and wind measurements, respectively in the lower and upper atmosphere. The narrowband sodium lidar, now capable of measuring mesopause region (80-110km in altitude) temperature and wind on 24-hour continuous basis, has enjoyed considerable success with continued NSF funding for upper atmospheric research since 1989. Its technology and innovations were and are beingduplicated by National and International Middle and Upper Atmospheric Research Facilities. In the past ten years, Prof. She’s research has been supported by NSF, NASA and AFOSR.Research PublicationsProfessor She has co-authored about 170 papers in refereed journals. A selected list since 1998:109. She, C. Y. and U. von Zahn, The concept of two-level mesopause: Support through new lidar observation, J.Geophys. Res., 103, 5855 - 5863, 1998.110. She, C. Y., S. W. Thiel and D. A. Krueger, Observed episodic warming at 86 and 100 km between 1990 and 1997: Effects of Mount Pinatubo eruption, Geophys. Res. Lett., 25, 497 - 500, 1998.114. She, C. Y., and R. P. Lowe, Seasonal temperature variations in the mesopause region at mid-latitude: comparison of lidar and hydroxyl rotational temperatures using WINDII/UARD OH height profiles, J. Atmo.Solar-Terr. Physics, 60, 1573-1583, 1998.119. She, C. Y., S. S. Chen, Z. L. Hu, J. Sherman, J. D. Vance, V. Vasoli, M. A. White, J. R. Yu, and D. A.Krueger, Eight-year climatology of nocturnal temperature and sodium density in the mesopause region (80 to 105 km) over Fort Collins, CO (41o N, 105o W), Geophys. Res. Lett., 27, 3289 - 3292, 2000.120. She, Chiao-Yao, Spectral structure of laser light scattering revisited: bandwidths of nonresonant scattering lidar, Appl. Opt. 40, 4875-4884, 2001.124. She, C. Y., Joe D. Vance, B. P. Williams, D. A. Krueger, H. Moosuller, D. Gibson-Wilde, and D. C. Fritts, Lidar studies of atmospheric dynamics near polar mesopause, EOS, Transactions, American Geophysical Union, 83 (27), P.289 and P.293, 2002.125. She, C. Y., Songsheng Chen, B. P. Williams, Zhilin Hu, David A. Krueger and M. E. Hagan, Tides in the mesopause region over Fort Collins, CO (41o N, 105o W) based on lidar temperature observations coveringfull diurnal cycles, Jour. Geophys. Research, 107, N. 0, 10.1029/2001JD001189, 2002.134. She, C. Y., Jim Sherman, Tao Yuan, B. P. Williams, Kam Arnold, T. D.Kawahara, Tao Li, LiFang Xu, J. D.Vance and David A. Krueger, The first 80-hour continuous lidar campaign for simultaneous observation of mesopause region temperature and wind, Geophys. Res. Lett.30, 6, 52, 10.1029/2002GL016412, 2003.136. She, C. Y., and D. A. Krueger, Impact of natural variability in the 11-year mesopause region temp erature observation over Fort Collins, CO (41N, 105W), Adv. Space Phys. 34, 330-336, 2004.137. Beig, G.; Keckhut, P.; Lowe, R. P.; Roble, R. G.; Mlynczak, M. G.; Scheer, J.; Fomichev, V. I.; Offermann,D.; French, W. J. R.; Shepherd, M. G.; Semenov, A. I.; Remsberg,E. E.; She, C. Y.; Lübken,F. J.; Bremer, J.;Clemesha, B. R.; Stegman, J.; Sigernes, F.; Fadnavis, S. (2003), Review of mesospheric temperature trends,Rev. Geophys., Vol. 41, No. 4, 1015, 10.1029/2002RG000121.138.Chiao-Yao She, Initial full-diurnal-cycle mesopause region lidar observations: Diurnal-means and tidal perturbations of temperature and winds over Fort Collins, CO (41N, 105W), PSMOS 2002, J. Atmo. Solar-Terr. Phys. 66, 663-674, 2004.139. She, C. Y.,Tao Li, Biff P. Williams, Tao Yuan and R. H. Picard (2004), Concurrent OH imager and sodium temperature/wind lidar observation of a mesopause region undular bore event over Fort Collins/Platteville, CO, J. Geophys. Res. 109, D22107, doi:10.1029/2004JD004742.140. She, C.Y., T. Li, R. C. Collins, T. Yuan, B. P. Williams, T. D. Kawahara, J. D. Vance, P. Acott, D. A. Krueger,H.-L. Liu, and M. E. Hagan (2004), Tidal perturbations and variability in the mesopause region over FortCollins, CO (41N, 105W): Continuous multi-day temperature and wind lidar observations, Geophys. Res. Lett., 31, L24111, doi:10.1029/2004GL021165.142. Fritts, D. C., B. P. Williams, C. Y. She, J. D. Vance, r. Rapp, F.-J. L¨ubken, A. F. J. Schmidlin, A. M¨ullemann R. A. Goldberg (2004) Observations of extreme temperature and wind gradients near the summer mesopause during the MaCWAVE/MIDAS rocket campaign, Geophys. Res. Lett., 31, L24S06, doi:10.1029/2003GL019389.144. Tao Li, C. Y. She, Bifford P. Williams, Tao Yuan, Richard L. Collins, Lois M. Kieffaber and Alan W.Peterson (2005), Concurrent OH imager and sodium temperature/wind lidar observation of localized ripples over Northern Colorado, J. Geophys. Res. 110, D13110, doi:10.1029/2004JD004885146. Chiao-Yao She (2005), On atmospheric lidar performance comparison: from power aperture to power–aperture–mixing ratio–scattering cross-section, Modern Optics, 52, 2723-2729, DOI:10.1080/09500340500352618.147. She, C. Y., B. P. Williams, P. Hoffmann, R. Latteck, G. Baumgarten, J. D. Vance, J. Fiedler, P. Acott, D. C.Fritts, F.-J. Luebken (2006): Observation of anti-correlation between sodium atoms and PMSE/NLC in summer mesopause at ALOMAR, Norway (69N, 12E), J. Atmos. Solar-Terres. Phys. 68, 93-101.148. Yuan, T., C. Y. She, M. E. Hagan, B. P. Williams, T. Li, K. Arnold, T. D. Kawahara, P. E. Acott, J. D. Vance,D. A. Krueger and R. G. Roble (2006), Seasonal variation of diurnal perturbations in mesopause-regiontemperature, zonal, and meridional winds above Fort Collins, CO (40.6°N, 105°W), J. Geophys. Res. 111, D06103, doi:10.1029/2004JD005486.149. Xu, Jiyao, C. Y. She, Wei Yuan, C. J. Mertens, Marty Mlynzack, J. R. Russell (2006), Comparison between the temperature measurements by TIMED/SABER and Lidar in the midlatitude, J. Geophys. Res., 111, A10S09, doi:10.1029/2005JA011439.150. Xu, J., A. K. Smith, R. L. Collins, and C.-Y. She (2006), Signature of an overturning gravity wave in the mesospheric sodium layer: Comparison of a nonlinear photochemical-dynamical model and lidar observations, J. Geophys. Res., 111, D17301, doi:10.1029/2005JD006749.151. Sherman, J. P., and C.-Y. She (2006), Seasonal variation of mesopause region wind shears, convective and dynamic instabilities above Fort Collins, CO: A statistical study, J. Atmo. Solar-Terr. Physics, 68, 1061-1074. 154. Davis, D. S., P. Hickson, G. Harriot and C. Y. She (2006), Temporal variability of the telluric sodium layer, Optics Lett. 31, 3369-3371.155. She, Chiao-Yao, J. D. Vance, T. D. Kawahara, B. P. Williams, and Q. Wu (2007), A proposed all-solid-state transportable narrowband sodium lidar for mesopause region temperature and horizontal wind measurements, Canadian Journal of Physics, 85, 111 – 118.156. Gumbeli, J., Z. Y. Fan, T. Waldemarsson, J. Stegman, G. Witts, E. J. Llewellyn, C.-Y. She and J. M. C. Plane (2007), Retrieval of global mesospheric sodium densities from the Odin satellite, Geophys. Res. Lett.,. 34, L04813, doi:10.1029/2006GL028687.157. Li, T., C.-Y. She, H.-L. Liu, and M. T. Montgomery (2007), Evidence of a gravity wave breaking event and the estimation of the wave characteristics from sodium lidar observation over Fort Collins, CO (41_N, 105_W) , Geophys. Res. Lett.,. 34, L05815, doi:10.1029/2006GL028988.158. She, C.-Y., J. Yue and Z.-A. Yan, J. W. Hair, J.-J. Guo, S.-H. Wu and Z.-S. Liu (2007), Direct-detection Doppler wind measurements with a Cabannes-Mie lidar: A. Comparison between iodine vapor filter and Fabry-Perot interferometer methods, Applied Optics 46, 4434-4443.159. She, C.-Y., J. Yue and Z.-A. Yan, J. W. Hair, J.-J. Guo, S.-H. Wu and Z.-S. Liu (2007), Direct-detection Doppler wind measurements with a Cabannes-Mie lidar:B. Impact of aerosol variation on iodine vapor filter methods, Applied Optics 46, 4444-4454.161. Tao Li, C.-Y. She, Scott E. Palo, Qian Wu, Han-Li Liu, and Murry L. Salby (2008), Coordinated Lidar and TIMED observations of the quasi-two-day wave during August 2002-2004 and possible quasi-biennial oscillation influence, Advanced Space Research 41, 1462-1470.162. Liu, H.-L., T. Li, C.-Y. She, J. Oberheide, Q. Wu, M. E. Hagan, J. Xu, R. G. Roble, M. G. Mlynczak, and J. M.Russell III (2007), Comparative study of short term diurnal tidal variability, J. Geophys. Res.(In press).163. She, Chiao-Yao, and David A. Krueger (2007), Laser-Induced Fluorescence: Spectroscopy in the Sky, Optics & Photonic News (OPN), 18(9), 35-41.164. Li, T., C. -Y. She, H.-L. Liu, T. Leblanc, and I. S. McDermid, Sodium lidar observed strong inertia-gravity wave activities in the mesopause region over Fort Collins, CO (41°N, 105°W) J. Geophys.Res. 112, D22104, doi:10.1029/2007JD008681.165. Yuan T., C.-Y. She and D. A. Krueger, F. Sassi, R. Garcia, R. Roble, H.-L. Liu, and H. Schmidt (2008), Climatology of mesopause region temperature, zonal wind and meridional wind over Fort Collins, CO (41ºN, 105ºW) and comparison with model simulations, J. Geophys. Res. 113, D03105, doi:10.1029/2007JD008697. 166. Liguo Su, R. J. Coolins, D. A. Krueger and C.-Y. She, Statistical Analysis of Sodium Doppler Wind-Temperature Lidar Measurements of Vertical Heat Flux, J. Atm. Oceanic Tech. (In press).167. Yuan, T., C. Y. She, Hauke Schmidt, David A. Krueger, Steven Reising, Seasonal variations of semidiurnal tidal-period perturbations in mesopause region temperature, zonal and meridional winds above Fort Collins, CO(40.6°N, 105°W), J. Geophys. (In oress).168. Yue, J., S. L. Vadas2, C-Y She, et al. (2008), A study of OH imager observed concentric gravity waves near Fort Collins on May 11, 2004, Geophys. Res. Lett. (submitted).169. Vadas, S. L., J. Yue, C.-Y. She and P. Stamus (2008), The effects of winds on concentric rings of gravity waves from a thunderstorm near Fort Collins in May 2004, J. Geophys.Res., (submitted).。